Analytical Model of Wave Loads and Motion Responses for a ... · to the wave induced hydrodynamic...
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Research Article Analytical Model of Wave Loads and Motion Responses for a Floating Breakwater System with Attached Dual Porous Side Walls Weiliang Qiao , 1 Keh-Han Wang, 2 Wenqi Duan, 1 and Yuqing Sun 1 Marine Engineering College, Dalian Maritime University, Dalian , China Dept. of Civil and Environmental Engineering, Univ. of Houston, Houston, TX , USA Correspondence should be addressed to Weiliang Qiao; qiao [email protected] Received 17 September 2018; Accepted 6 November 2018; Published 19 November 2018 Academic Editor: Roman Lewandowski Copyright © 2018 Weiliang Qiao et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A set of two-dimensional analytical solutions considering the effects of diffraction and radiation are presented in this study to investigate the hydrodynamic interaction between an incident linear wave and a proposed floating breakwater system consisting of a rectangular-shaped body and two attached vertical side porous walls in an infinite fluid domain with finite water depth. e Matched Eigenfunction Expansion Method (MEEM) for multiple fluid domains is applied to derive theoretically the velocity potentials and associated unknown coefficients for wave diffraction and body motion induced radiation in each subdomain. Also, the exciting forces, as well as the added mass and damping coefficients for the floating breakwater system under the surge, heave, and pitching motions, are formulated. e displacements of breakwater motions are determined by solving the equation of motion. As a verification of the analytical model, the present solutions of the limiting cases in terms of exciting forces, moments, added masses, and damping coefficients are found to be well matched with other published numerical results. Additionally, the hydrodynamic performances and the dynamic responses in terms of Response Amplitude Operators (RAOs) of the proposed floating breakwater system are evaluated versus various dimensionless variables, such as wavelength and porous-effect parameter. e results show that the attached porous walls with selected porous properties are observed to have the advantages of reducing wave impacts on the floating breakwater system and at the same time its dynamic responses are also noticeably improved. 1. Introduction e ocean-oriented activities, such as oil and gas extrac- tion, aquacultural production, and recreations, have been identified to generate tremendous values to humans. Pro- tection of the shoreline structures or offshore production facilities by degradation of the wave-structure interaction is therefore a vital issue in the coastal and offshore engi- neering applications. Various artificial structures have been designed and optimized for this purpose, of which the floating breakwaters or barriers are considered as valuable alternatives to conventional fixed gravity-type breakwaters [1]. e performance optimization of a breakwater, especially the type of floating breakwater, has received much attention with the advantage of energy dissipation [2]. Additionally, the porous structures may provide an added alternative to improve the performance of floating breakwaters. Hydrodynamic analysis is widely applied to examine the wave-reduction performance of a floating breakwater. Mei and Black [3] investigated the wave forces acting on the rectangular floating obstacles by the variational-formulation- based numerical approach. Later, the wave radiation induced by small oscillations of a floating body was also studied [4]. A simplified analytical model to estimate the hydrodynamic performance of a long rectangular floating breakwater was developed under the assumption of a small gap condition [5]. Also, the diffraction and radiation problem of a rectangular buoy were analyzed under the conditions of heave oscillation [6], as well as in three dimensional scale, the sway, heave, and rolling motions [7]. Damping mechanisms of a floating breakwater were investigated numerically and experimentally considering the cases of a regular pontoon, a regular pontoon with wing plates attached, and a regular pontoon with wing plates penetrated [8]. Additionally, the hydrodynamic Hindawi Mathematical Problems in Engineering Volume 2018, Article ID 1295986, 14 pages https://doi.org/10.1155/2018/1295986
Transcript of Analytical Model of Wave Loads and Motion Responses for a ... · to the wave induced hydrodynamic...
Research ArticleAnalytical Model of Wave Loads and MotionResponses for a Floating Breakwater Systemwith Attached Dual Porous Side Walls
Weiliang Qiao 1 Keh-HanWang2 Wenqi Duan1 and Yuqing Sun1
1Marine Engineering College Dalian Maritime University Dalian 116026 China2Dept of Civil and Environmental Engineering Univ of Houston Houston TX 77024 USA
Correspondence should be addressed to Weiliang Qiao qiao wlhotmailcom
Received 17 September 2018 Accepted 6 November 2018 Published 19 November 2018
Academic Editor Roman Lewandowski
Copyright copy 2018 Weiliang Qiao et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
A set of two-dimensional analytical solutions considering the effects of diffraction and radiation are presented in this study toinvestigate the hydrodynamic interaction between an incident linear wave and a proposed floating breakwater system consistingof a rectangular-shaped body and two attached vertical side porous walls in an infinite fluid domain with finite water depthThe Matched Eigenfunction Expansion Method (MEEM) for multiple fluid domains is applied to derive theoretically the velocitypotentials and associated unknown coefficients for wave diffraction and body motion induced radiation in each subdomain Alsothe exciting forces aswell as the addedmass anddamping coefficients for the floating breakwater systemunder the surge heave andpitching motions are formulated The displacements of breakwater motions are determined by solving the equation of motion Asa verification of the analytical model the present solutions of the limiting cases in terms of exciting forces moments added massesand damping coefficients are found to be well matched with other published numerical results Additionally the hydrodynamicperformances and the dynamic responses in terms of Response Amplitude Operators (RAOs) of the proposed floating breakwatersystem are evaluated versus various dimensionless variables such as wavelength and porous-effect parameterThe results show thatthe attached porous walls with selected porous properties are observed to have the advantages of reducing wave impacts on thefloating breakwater system and at the same time its dynamic responses are also noticeably improved
1 Introduction
The ocean-oriented activities such as oil and gas extrac-tion aquacultural production and recreations have beenidentified to generate tremendous values to humans Pro-tection of the shoreline structures or offshore productionfacilities by degradation of the wave-structure interactionis therefore a vital issue in the coastal and offshore engi-neering applications Various artificial structures have beendesigned and optimized for this purpose of which thefloating breakwaters or barriers are considered as valuablealternatives to conventional fixed gravity-type breakwaters[1] The performance optimization of a breakwater especiallythe type of floating breakwater has received much attentionwith the advantage of energy dissipation [2] Additionallythe porous structures may provide an added alternative toimprove the performance of floating breakwaters
Hydrodynamic analysis is widely applied to examine thewave-reduction performance of a floating breakwater Meiand Black [3] investigated the wave forces acting on therectangular floating obstacles by the variational-formulation-based numerical approach Later the wave radiation inducedby small oscillations of a floating body was also studied [4]A simplified analytical model to estimate the hydrodynamicperformance of a long rectangular floating breakwater wasdeveloped under the assumption of a small gap condition [5]Also the diffraction and radiation problem of a rectangularbuoy were analyzed under the conditions of heave oscillation[6] as well as in three dimensional scale the sway heaveand rolling motions [7] Damping mechanisms of a floatingbreakwaterwere investigated numerically and experimentallyconsidering the cases of a regular pontoon a regular pontoonwith wing plates attached and a regular pontoon withwing plates penetrated [8] Additionally the hydrodynamic
HindawiMathematical Problems in EngineeringVolume 2018 Article ID 1295986 14 pageshttpsdoiorg10115520181295986
2 Mathematical Problems in Engineering
performances of rectangular boxes of fixed single and fixeddouble subject to the heave motion were estimated under theactions of either regular or irregular water waves [9] Laterthe wave absorbing effectiveness of different types of floatingbreakwaters such as single box double boxes or board netwas also investigated [10]
Due to the advantage of energy dissipation the water-wave absorbing phenomena have been observed when wavespass through a porousmedium that is a porous structure caneffectively attenuate both reflected and transmitted wavesFollowing the porous wavemaker theory of Chwang [11] con-siderable studies related to the interaction between porousstructures and water waves were performed theoretically andexperimentally especially the porous barriers or the porousstructures attached to the floating breakwaters The waveenergy dissipation of thin permeable wall was verified byconducting sets of experiments [12] Analytical approacheshave been carried out by Manam and Sivanesan [13] andChwang and Dong [14] as well as Zhu and Chwang [15] tostudy the reflection and transmission problem of water wavespropagating past vertical porous barriers The hydroelasticissues between gravity waves and a submerged horizontalflexible porous plate were investigated by Behera and Sahoo[16] The studies of perforated wall breakwaters and dualsubmerged horizontal porous plates were given in [17 18]respectively In order to examine the effectiveness of waveabsorption by porous plates the interactions between dif-ferent types of porous plates and oblique monochromaticwaves were investigated by Cho and Kim [19] The conceptof porous barrier was extended by Wang and Ren [20 21]to analytically study the interaction between water wavesand a flexible porous breakwater or in a three-dimensionaldomain a concentric porous cylinder systemThe diffractionproblems of either a circular porous plate submerged hori-zontally or a system with dual circular porous plates wereinvestigated based on the linear potential flow theory [2223] Recently the scattering of water waves by a rectangularfloating body that attached with two vertically porous wallswas investigated analytically and experimentally by Qiao etal [24] In practices the types of porous structure are variousdepending on the requirements of engineering problemhowever single or several vertical porous plates (includingbarrier and circular) and porous cylinder are the mostcommon in engineering fieldThe vertical porous plates wereoften employed to absorb waves to protect coastal [25 26]furthermore a set of porous plates were placed at the bottomof a wave tank to absorb waves [27]The porous cylinder wasemployed to protect structure in Trafalgar offshore windfarm[28] almost for the same purpose a concentric two-cylindersystem involving an interior impermeable cylinder andexternal porous cylinder was designed for Ekofisk gravityoffshore structure in the North Sea [29] then the conceptof single-layered porous cylinder was extended to double-layered perforated cylinder by Xiao et al [30] and theexperimentswere conducted to investigate the hydrodynamicperformances
To achieve the combination of the advantages of a floatingbody and porous wall features a new breakwater systemthat consists of a partially submerged pontoon-like body
and two attached porous side walls is proposed in thisstudy to provide a protected region sheltering from waveaction The results of wave reflectiontransmission and waveinduced hydrodynamic loads under the condition that thebreakwater system is in a fixed position were given in Qiaoet al [24] The investigation of wave interactions with thebreakwater system is further extended to include the effectof the induced motions of the floating breakwater systemthe so-called radiation problem Under the assumptions oflinear incident waves and small translational and rotationalmotions of surge heave and pitch this study focuses on theinvestigation of hydrodynamic coefficients such as the addedmass and damping coefficient and the dynamic responsesof the floating breakwater system through developing ananalytical model with solutions of diffracted and radiatedwaves and the equation of motion which is formulated andsolved The solutions of the velocity potentials in the definedregions including the diffraction and radiation velocitypotentials are obtained by solving the formulated governingequations as well as the boundary and matching conditionsTo verify the analytical solutions derived from this studysome limited numerical results published in literatures wereadopted for comparisons Additionally the effects of theincident wave parameters and porous walls conditions on thehydrodynamic coefficients and RAOs (Response AmplitudeOperators) are examined and discussed
2 Theoretical Formulations and Solutions
The problem statement as illustrated in Figure 1 showsa train of right-going small-amplitude Airy waves withthe frequency 120596 wave height H and wave length 120582 tointeract normally with a floating breakwater system thatconsists of a partially submerged rectangular structure andtwo attached fully submerged side porous walls Subjectto the wave induced hydrodynamic loads the breakwatersystem responds with two translational motions and onerotational motion ie surge heave and pitching motionsThe Cartesian coordinates system is employed to definethis two-dimensional boundary value problem The x-axisis along the still water surface where 119909 = 0 is set at thecenter of the floating breakwater system and the z-axis pointsvertically upwards with an origin located at the still watersurface The rectangular structure with a length of 2a anda submergence of s is initially positioned in a domain withundisturbed constant water depth h Both of the thin porouswalls have a length of d-s
In this study the fluid is assumed to be incompressibleand inviscid and the motion irrational Subsequently thepotential flow theory providing that the velocity potentialsatisfies the Laplace Equation with the defined boundaryconditions could be applied to solve thewaves and breakwaterinteraction problem As shown in Figure 1 the entire fluiddomain is divided into three subdomains namely regions III and III respectively Region I is the subdomain upstreamof the breakwater system (119909 le 119886 minusℎ le 119911 le 0) region II is thesubdomain between the two porous walls (minus119886 le 119909 le 119886 minus119889 le119911 le minus119904) and region III is the subdomain downstream of thebreakwater system (119886 le 119909 minusℎ le 119911 le 0) In the fluid domain
Mathematical Problems in Engineering 3
X
Z
sd
h
Porous wall
2a
surface
bottomI II III
Figure 1 Schematic diagram of a floating body with attached two porous walls under the action of an incident wave and forced surge heaveand pitching motions
the Laplace equation of a complete velocity potential servesas the basic equation to describe the fluid motion namely
nabla2120601 = 0 (1)
The fluid velocity potential 120601 includes three componentsincident wave potential 120601119868 diffracted wave potential 120601119863 andthe radiated wave potential Φ119877 which covers the inducedmotions of the floating breakwater system In this studythe dynamic motions of surge heave and pitching areconsidered Thus the velocity potential 120601 can be expressedas
120601 = 120601119868 + 120601119863 + 3sum119895=1
Φ(119871)119877 (2)
where the subscript L = 1 2 3 donates the surge heaveand pitching motions of the breakwater system respectivelyIn this study the incident wave potential is given as
120601119868 = minusi1198921198672120596 119890i1198960(119909+119886) cosh 1198960 (119911 + ℎ)cosh 1198960ℎ 119890minusi120596119905 (3)
where 1198960 is the wave number120596 is the wave frequency and119892 is the gravitational acceleration Here 1198960 and 120596 satisfy theusual dispersion relation
1205962 = 1198921198960 tanh (1198960ℎ) (4)
21 eoretical Formulations and Solutions of the Incidentand Diffracted Velocity Potentials To solve the wave radi-ation problem in a linear wave system the wave inducedhydrodynamic forces and moment according to the wavediffraction approach by assuming the body is in a fixedposition are required to be determined priorly The wavediffraction solutions for the proposed breakwater systemwithtwo attached porous walls have been obtained in a previousstudy given inQiao et al [24] and Yip and Chwang [31] Hereonly limited formulations and solutions are summarizedbelow Let us define the wave diffraction potentials as 120601119863119894119894 = 1 2 3 stands for the subdomains I II and III respectivelyThe Laplace equations of 120601119863119894 can be solved with the uses ofthe linearized kinematic and dynamic free surface boundarycondition bottom boundary condition no flow condition at
the body surface andmatching conditions at the interfaces ofporous wall locations including the porous flow conditionsexpressed as (Chwang [11])
1205971206011198631120597119909 = 1205971206011198632120597119909 = 1198871120583 (1199011198631 minus 1199011198632)119886119905 119909 = minus119886 minus119889 le 119911 le minus119904
(5)
1205971206011198633120597119909 = 1205971206011198632120597119909 = 1198872120583 (1199011198632 minus 1199011198633)119886119905 119909 = minus119886 minus119889 le 119911 le minus119904
(6)
where 1198871 and 1198872 having the dimension of length arerespectively material constants for the porous walls A andB 119901119863119894 (i=1 2 3) are the dynamic pressures induced bythe incident and diffracted velocity potentials and 120583 is thedynamic viscosity of the fluid The wave diffraction solutionsare derived as
1206011198631 = infinsum119898=0
[119868119898119890119896119898(119909+119886) + 119865119898119890minus119896119898(119909+119886)] cos 119898 (119911 + ℎ)cos 119898ℎ
sdot 119890minusi120596119905(7)
1206011198632 = (1198620119909 + 1198640) 119890minusi120596119905+ infinsum119898=1
[119862119898 cosh (119903119898119909)cosh (119903119898119886) + 119864119898 sinh (119903119898119909)
sinh (119903119898119886) ]sdot cos 119903119898 (119911 + ℎ) 119890minusi120596119905
(8)
1206011198633 = infinsumm=0
[119879119898119890119896119898(119909minus119886)] cos 119898 (119911 + ℎ)cos 119898ℎ 119890minusi120596119905 (9)
where 119865119898 119862119898 119864119898 and 119879119898 (119898 = 0 1 2 3 ) areunknown complex coefficients to be determined 119868119898 119898 and119903119898 (119898 = 0 1 2 3 ) are expressed separately as
119868119898 = minusi1198921198672120596 0 0 0 (10)
119898 = i1198960 minus1198961 minus1198962 minus1198963 minus119896119898 (11)
119903119898 = 119898120587ℎ minus 119904 (12)
4 Mathematical Problems in Engineering
Here 119896119898 can be calculated from
1205962 = minus119892119896119898 tanh (119896119898ℎ) (13)
Substituting (7) (8) and (9) into the following derivedmatching equations
int0minusℎ
1205971206011198631120597119909 cos 119899 (119911 + ℎ) 119889119911= intminus119904minusℎ
1205971206011198632120597119909 cos 119899 (119911 + ℎ) 119889119911119909 = minus119886 (119899 = 0 1 2 3 )
(14)
int0minusℎ
1205971206011198633120597119909 cos 119899 (119911 + ℎ) 119889119911= intminus119904minusℎ
1205971206011198632120597119909 cos 119899 (119911 + ℎ) 119889119911119909 = minus119886 (119899 = 0 1 2 3 )
(15)
intminus119889minusℎ
1206011198631 cos 119903119899 (119911 + ℎ) 119889119911 + intminus119904minus119889
1198631 cos 119903119899 (119911 + ℎ) 119889119911= intminus119904minusℎ
1206011198632 cos 119903119899 (119911 + ℎ) 119889119911119909 = minus119886 (119899 = 0 1 2 )
(16)
intminus119889minusℎ
1206011198633 cos 119903119899 (119911 + ℎ) 119889119911 + intminus119904minus119889
1198633 cos 119903119899 (119911 + ℎ) 119889119911= intminus119904minusℎ
1206011198632 cos 119903119899 (119911 + ℎ) 119889119911119909 = minus119886 (119899 = 0 1 2 )
(17)
and utilizing the orthogonality properties of solutioneigenfunctions the unknown coefficients 119862119899 119863119899 119877119899 119879119899 (119899 =0 1 2 3 119872) for the diffraction potentials can be deter-mined by solving the 4(M+1) system of algebraic equationsHere the terms with summation of infinite series are approx-imated by a large but finite number of M terms In (16) and(17) 1198631 and 1198633 are given as
1198631 = infinsum119898=0
[(119868119898 + 119865119898) minus 119898i119866011198960 (119868119898 minus 119865119898)]
sdot cos 119898 (119911 + ℎ)cos 119898ℎ
(18)
1198633 = infinsum119898=0
(1 + 119898i119866011198960) 119879119898 cos 119898 (119911 + ℎ)
cos 119898ℎ (19)
where 11986601 = 1205881205961198871(1205831198960) and 11986602 = 1205881205961198872(1205831198960) aredimensionless parameters defined according to Chwang [11]to represent the porous effect parameters of porous walls Aand B respectively
22 eoretical Formulations of the Radiated Velocity Poten-tials For the radiation problem the displacements of thefloating breakwater system in motion are given as
119883119895 = 1198830119895119890minusi120596119905 119895 = 1 2 3 (20)
where 119883119900119895 (119895 = 1 2 3) are the amplitudes of surge heaveand pitching motions respectively Let 120601(119871)119877119894 (119871 = 1 2 3 119894 =1 2 3) be the spatial radiated potentials describing the small-amplitude oscillations where 119871119905ℎ indices denote respectivelythe induced surge heave and pitching motions while the 119894119905ℎindices represent the subdomains of I II and III respectivelyThen the total radiated potentials in any of the flow domainscan be expressed as
Φ(119871)119877 = 119883119895120601(119871)119877 = Re [minusi1205961198830119895120601(119871)119877 (119909 119911) 119890minusi120596119905] (21)
As the velocity potentials120601(119871)119877 satisfy the Laplace equationwe have
nabla2120601(119871)119877119894 = 0 (119871 = 1 2 3 119894 = 1 2 3) (22)
The boundary conditions including the free surfaceboundary conditions bottom boundary conditions and thebody surface conditions for the surge heave and pitchingoscillations are listed as follows respectively
120597120601(1)119877119894120597119911 = 0 (119894 = 1 2 3) 119911 = ℎ1205972120601(1)1198771198941205971199052 + 120597120601(1)119877119894120597119911 = 0 (119894 = 1 3) 119911 = 0 |119909| ge 119886
120597120601(1)1198772120597119911 = 0 119911 = minus119904 |119909| le 119886(23)
120597120601(2)119877119894120597119911 = 0 (119894 = 1 2 3) 119911 = ℎ1205972120601(2)1198771198941205971199052 + 119892 120597120601(2)119877119894120597119911 = 0 (119894 = 1 3) 119911 = 0 |119909| ge 119886
120597120601(2)1198772120597119911 = 1 119911 = minus119904 |119909| le 119886(24)
120597120601(3)119877119894120597119911 = 0 (119894 = 1 2 3) 119911 = ℎ1205972120601(3)1198771198941205971199052 + 119892 120597120601(1)119877119894120597119911 = 0 (119894 = 1 3) 119911 = 0 |119909| ge 119886
120597120601(3)119877119894120597119899 = i120596 [(119911 minus 120585119911) 119899119909 minus (119909 minus 120585119909) 119899119911]119900119899 119887119900119889119910 119904119906119903119891119886119888119890
(25)
where 119899 = (119899119909 119899119911) is the unit normal vector and (120585119909 120585119911)is the rotational center of the pitching oscillation With theassumption that the rotational center is along the z axis wecan get 120585119909 = 0
Mathematical Problems in Engineering 5
23 Analytical Solutions of the Radiated Velocity PotentialsSimilar to the derivation of the solutions of the diffractedvelocity potentials the application of the method of sepa-ration variables to the governing equation (22) under theboundary conditions (23) (24) and (25) yields the radiatedvelocity potentials as follows
120601(119871)1198771 = infinsum119898=0
119860(119871)119898 119890minus119896119898(119909+119886) cos 119898 (119911 + ℎ)cos 119898ℎ 119890minusi120596119905 (26)
120601(1)1198772 = (119862(1)0 119909 + 119864(1)0 ) 119890minusi120596119905+ infinsum119898=1
[119862(1)119898 cosh (119903119898119909)cosh (119903119886119909) + 119864(1)119898 sinh (119903119898119909)
sinh (119903119886119909) ]sdot cos 119903119898 (119911 + ℎ) 119890minusi120596119905
(27)
120601(2)1198772 = (119911 + ℎ)2 minus 11990922 (ℎ minus 119904) + (119862(2)0 119909 + 119864(2)0 ) 119890minusi120596119905
+ infinsum119898=1
[119862(2)119898 cosh (119903119898119909)cosh (119903119886119909) + 119864(2)119898 sinh (119903119898119909)
sinh (119903119886119909) ]sdot cos 119903119898 (119911 + ℎ) 119890minusi120596119905
(28)
120601(3)1198772 = infinsum119898=1
120576 cos [120574119898 (119909 minus 119886)] cosh [120574119898 (119911 + ℎ)]+ (119862(3)0 119909 + 119864(3)0 ) 119890minusi120596119905+ infinsum119898=1
[119862(3)119898 cosh (119903119898119909)cosh (119903119886119909) + 119864(3)119898 sinh (119903119898119909)
sinh (119903119886119909) ]sdot cos 119903119898 (119911 + ℎ) 119890minusi120596119905
(29)
120601(119871)1198773 = infinsum119898=0
119861(119871)119898 119890119896119898(119909minus119886) cos 119898 (119911 + ℎ)cos 119898ℎ 119890minusi120596119905 (30)
where
120574119898 = 1198981205872119886 (119898 = 1 2 3 ) (31)
Combining the floating body surface condition at 119911 = minus119904minus119886 le 119909 le 119886 and the orthogonality of cos[120574119898(119909 minus 119886)] over theintegral interval of minus119886 le 119909 le 119886 the expression of 120576 can bederived as
120576 = cos 2119886120574119898sinh [120574119898 (ℎ minus 119904)] 1198861205741198983 (32)
24 Determination of theUnknownCoefficientsatAppearedin the Solutions of Radiated Velocity Potentials The solutionsof the unknown coefficients that appeared in the radiationpotentials can be again obtained by utilizing the matchingconditions at 119909 = plusmn119886 under the principle of continuouspressure and normal velocity We have the following
(a) Matching Conditions for the Surge-Induced Potential
120601(1)1198772 = 120601(1)1198771 minusℎ le 119911 le minus119889 119909 = minus119886120601(1)1198773 minusℎ le 119911 le minus119889 119909 = 119886 (33a)
120597120601(1)1198771120597119909 = 120597120601(1)1198772120597119909 minusℎ le 119911 le minus119889 119909 = minus1198861 minus119904 le 119911 le 0 119909 = 119886 (33b)
120597120601(1)1198773120597119909 = 120597120601(1)1198772120597119909 minusℎ le 119911 le minus119889 119909 = minus1198861 minus119904 le 119911 le 0 119909 = 119886 (33c)
(b) Matching Conditions for the Heave-Induced Potential
120601(2)1198772 = 120601(2)1198771 minusℎ le 119911 le minus119889 119909 = minus119886120601(2)1198773 minusℎ le 119911 le minus119889 119909 = 119886 (34a)
120597120601(2)1198771120597119909 = 120597120601(2)1198772120597119909 minusℎ le 119911 le minus119889 119909 = minus1198861 minus119904 le 119911 le 0 119909 = 119886 (34b)
120597120601(2)1198773120597119909 = 120597120601(2)1198772120597119909 minusℎ le 119911 le minus119889 119909 = minus1198861 minus119904 le 119911 le 0 119909 = 119886 (34c)
(c) Matching Conditions for the Pitching-Induced Potential
120601(3)1198772 = 120601(3)1198771 minusℎ le 119911 le minus119889 119909 = minus119886120601(3)1198773 minusℎ le 119911 le minus119889 119909 = 119886 (35a)
120597120601(3)1198771120597119909 = 120597120601(3)1198772120597119909 minusℎ le 119911 le minus119889 119909 = minus119886119911 minus 120585119911 minus119904 le 119911 le 0 119909 = 119886 (35b)
120597120601(3)1198773120597119909 = 120597120601(3)1198772120597119909 minusℎ le 119911 le minus119889 119909 = minus119886119911 minus 120585119911 minus119904 le 119911 le 0 119909 = 119886 (35c)
For the porous walls A and B at the boundaries 119909 =plusmn119886 the condition of continuity of normal velocity for flowspassing through the porous walls A and B leads to
120597120601(119871)1198771120597119909 = 120597120601(119871)1198772120597119909 = 119882119860 (119911 119905)119871 = (1 2 3) minus ℎ le 119911 le minus119889 119909 = minus119886
(36a)
120597120601(119871)1198773120597119909 = 120597120601(119871)1198772120597119909 = 119882119861 (119911 119905)119871 = (1 2 3) minus ℎ le 119911 le minus119889 119909 = minus119886
(36b)
where the porous flow velocities 119882119860(119911 119905) and 119882119861(119911 119905) areassumed to obey Darcys law that is they are linearly pro-portional to the pressure difference induced by the radiated
6 Mathematical Problems in Engineering
motions across their corresponding porous wall (Chwang[11]) With the application of the linearized Bernoulli equa-tion we have
119882119860 (119911 119905) = minus11986601 [ 1198960120596 (120597120601(119871)1198771120597119905 minus 120597120601(119871)1198772120597119905 )](119871 = 1 2 3) 119909 = minus119886 minus119889 le 119911 le minus119904
(37a)
119882119861 (119911 119905) = minus11986602 [ 1198960120596 (120597120601(119871)1198772120597119905 minus 120597120601(119871)1198773120597119905 )](119871 = 1 2 3) 119909 = 119886 minus119889 le 119911 le minus119904
(37b)
Utilizing the orthogonality property of eigenfunctionscos 119899(119911 + ℎ) for 120601(119871)1198771 and 120601(119871)1198773 (119871 = 1 2 3) over the integralinterval of minusℎ le 119911 le 0 the following equations can beformulated by the matching conditions of (33b) (33c) (34b)(34c) (35b) and (35c) as
int0minusℎ
120597120601(1)1198771120597119909 cos 119899 (119911 + ℎ) 119889119911= intminus119904minusℎ
120597120601(1)1198772120597119909 cos 119899 (119911 + ℎ) 119889119911+ int0minus119904
1 cos 119899 (119911 + ℎ) 119889119911 (119909 = minus119886)(38)
int0minusℎ
120597120601(1)1198773120597119909 cos 119899 (119911 + ℎ) 119889119911= intminus119904minusℎ
120597120601(1)1198772120597119909 cos 119899 (119911 + ℎ) 119889119911+ int0minus119904
1 cos 119899 (119911 + ℎ) 119889119911 (119909 = 119886)(39)
int0minusℎ
120597120601(2)1198771120597119909 cos 119899 (119911 + ℎ) 119889119911= intminus119904minusℎ
120597120601(2)1198772120597119909 cos 119899 (119911 + ℎ) 119889119911 (119909 = minus119886)(40)
int0minusℎ
120597120601(2)1198773120597119909 cos 119899 (119911 + ℎ) 119889119911= intminus119904minusℎ
120597120601(2)1198772120597119909 cos 119899 (119911 + ℎ) 119889119911 (119909 = 119886)(41)
int0minusℎ
120597120601(3)1198771120597119909 cos 119899 (119911 + ℎ) 119889119911= intminus119904minusℎ
120597120601(3)1198772120597119909 cos 119899 (119911 + ℎ) 119889119911+ int0minus119904
(119911 minus 120585119911) cos 119899 (119911 + ℎ) 119889119911 (119909 = minus119886)(42)
int0minusℎ
120597120601(3)1198773120597119909 cos 119899 (119911 + ℎ) 119889119911
= intminus119904minusℎ
120597120601(3)1198772120597119909 cos 119899 (119911 + ℎ) 119889119911+ int0minus119904
(119911 minus 120585119911) cos 119899 (119911 + ℎ) 119889119911 (119909 = 119886)
(43)
Additionally by combining (36a) (36b) (37a) and (37b)with the matching conditions given in (33a) (34a) and(35a) respectively and applying the orthogonality propertyof eigenfunctions cos 119903119899(119911 + ℎ) for 120601(119871)1198772 (L = 1 2 3) over theintegral interval of minusℎ le 119911 le minus119904 we have
intminus119904minusℎ
120601(119871)1198772 cos 119903119899 (119911 + ℎ) 119889119911= intminus119889minusℎ
120601(119871)1198771 cos 119903119899 (119911 + ℎ) 119889119911+ intminus119904minus119889
(119871)1198771 cos 119903119899 (119911 + ℎ) 119889119911(119909 = minus119886 119871 = 1 2 3)
(44)
intminus119904minusℎ
120601(119871)1198772 cos 119903119899 (119911 + ℎ) 119889119911= intminus119889minusℎ
120601(119871)1198773 cos 119903119899 (119911 + ℎ) 119889119911+ intminus119904minus119889
(119871)1198773 cos 119903119899 (119911 + ℎ) 119889119911(119909 = 119886 119871 = 1 2 3)
(45)
where
(1)1198771 = infinsum119898=0
119860(1)119898 (1 + 119898i119866011198960)
cos 119898 (119911 + ℎ)cos 119898ℎ
+ 1i119866011198960
(46)
(2)1198771 = infinsum119898=0
119860(2)119898 (1 + 119898i119866011198960)
cos 119898 (119911 + ℎ)cos 119898ℎ (47)
(3)1198771 = infinsum119898=0
119860(3)119898 (1 + 119898i119866011198960)
cos 119898 (119911 + ℎ)cos 119898ℎ
+ 119911i119866011198960
(48)
Mathematical Problems in Engineering 7
(1)1198773 = infinsum119898=0
119861(1)119898 (1 + 119898i119866021198960)
cos 119898 (119911 + ℎ)cos 119898ℎ
minus 1i119866021198960
(49)
(2)1198773 = infinsum119898=0
119861(2)119898 (1 minus 119898i119866021198960)
cos 119898 (119911 + ℎ)cos 119898ℎ (50)
(3)1198773 = infinsum119898=0
119861(3)119898 (1 + 119898i119866021198960)
cos 119898 (119911 + ℎ)cos 119898ℎ
minus 119911i119866021198960
(51)
Again the summation of terms with the infinite seriesis approximated by a large but finite numbers of termseg a total of M terms Then the unknown coefficients119860(1)119898 119862(1)119898 119864(1)119898 119861(1)119898 (119898 = 0 1 2 119872) [4(M+1) unknownstotally] can be determined by solving the 4(M+1) sys-tem of algebraic equations formulated in (38) (39) (44)and (45) (for L=1) Similarly the unknown coefficients119860(2)119898 119862(2)119898 119864(2)119898 119861(2)119898 (119898 = 0 1 2 119872) can be calculatedfrom the 4(M+1) system of algebraic equations expressedin (40) (41) (44) and (45) (for L=2) while the coefficients119860(3)119898 119862(3)119898 119864(3)119898 119861(3)119898 (119898 = 0 1 2 119872) can be evaluated bysolving the 4(M+1) system of algebraic equations formulatedin (42) (43) (44) and (45) (for L=3) After the determinationof the above unknown coefficients a series of engineeringproperties important to practical applications could be cal-culated for example the hydrodynamic coefficients and themotion induced dynamic responses of the breakwater system
3 Exciting Forces andHydrodynamic Coefficients
31 Exciting Forces To determine the hydrodynamic coeffi-cients related to the inducedmotions of the floating breakwa-ter system the exciting forces and moment of the combinedincident and diffracted waves on the breakwater need tobe computed as inputs for solving the equation of motionBy integrating the dynamic pressures with the use of thevelocity potentials of incident and diffracted waves (see (7)-(9)) over the wet surface the hydrodynamic forces alongthe x direction (1198651198631) and along the z direction (1198651198632) can beformulated respectively as (Qiao et al [24])
1198651198631= i120588120596 infinsum
119898=0
(119868119898 + 119865119898 minus 119879119898) [sin 119898ℎ minus sin 119898 (ℎ minus 119889)]119898 cos 119898ℎ
sdot 119890minusi120596119905 + i120588120596 lowast 2119886 (119889 minus 119904) 1198620119890minusi120596119905minus i120588120596 infinsum
119898=0
2119864119898119903119898 sin 119903119898 (ℎ minus 119889) 119890minusi120596119905(52)
1198651198632 = i120596120588 [21198861198640 + 2 infinsum119898=1
119862119898119903119898 tanh (119903119898119886) cos 119903119898 (ℎ minus 119904)]sdot 119890minusi120596119905
(53)
Also the wave exciting moment (1198651198633) that is referencedto a rotational center (0120585119911) can be derived as
1198651198633 = i120588120596
(119868119898 + 119865119898 minus 119879119898)
sdot [[[[[[
119889 sin 119898 (ℎ minus 119889) minus 120585119911 sin 119898ℎ + 120585119911 sin 119898 (ℎ minus 119889)119898 cos 119898ℎ
+cos 119898ℎ minus cos 119898 (ℎ minus 119889)1198982 cos 119898ℎ
]]]]]]
sdot 119890minusi120596119905 + i120588120596 1198861198620 (1198892 minus 1199042)
minus infinsum119898=1
2119864119898119903119898 [119889 sin 119903119898 (ℎ minus 119889)
+ cos 119903119898 (ℎ minus 119904) minus cos 119903119898 (ℎ minus 119889)119903119898 ] 119890minusi120596119905
+ i120588120596 [21198861198620120585119911 (119889 minus 119904) minus infinsum119898=1
2119864119898120585119911119903119898 sin 119903119898 (ℎ minus 119889)]
sdot 119890minusi120596119905 minus i120588120596 2119862011988633 + infinsum119898=1
[ 2119886119864119898 cos 119903119898 (ℎ minus 119904)119903119898 tanh (119903119898119886)minus 2119864119898 cos 119903119898 (ℎ minus 119904)1199031198982 ] 119890minusi120596119905
(54)
32 Hydrodynamic Coefficients The radiation forces inducedby the translational and rotational motions of the floatingbreakwater system can be calculated from the derived radi-ation potentials utilizing the linearized Bernoulli equation as
119865119871119896 = i120588120596 intΓ
Φ(119871)119877119896 119890minusi120596119905119899119895119889Γ= 12058812059621198830119895 int
Γ120601(119871)119877119896 119890minusi120596119905119899119895119889Γ
= 12059621198830119895119898119871119896119890minusi120596119905 + i1205961198830119895119889119871119896119890minusi120596119905(55)
where 119898119871119896 is the radiation-induced added mass and the119889119871119896 denotes the radiation damping The expressions arerespectively
119898119871119896 = Re(120588 intΓ
120601(119871)119877119896119899119895119889Γ) = Re (120588119871119896) (56)
119889119871119896 = Im (120588120596 intΓ
120601(119871)119877119896119899119895119889Γ) = Im (120588120596119871119896) (57)
8 Mathematical Problems in Engineering
119871119896 can be computed with formulations given below
11 = (int0minus119889
120601(1)1198771 10038161003816100381610038161003816119909=minus119886 119889119911 minus int0minus119889
120601(1)1198773 10038161003816100381610038161003816119909=119886 119889119911)+ (intminus119904minus119889
120601(1)1198772 10038161003816100381610038161003816119909=minus119886 119889119911 minus intminus119904minus119889
120601(1)1198772 10038161003816100381610038161003816119909=119886 119889119911)= 119872sum119898=1
119860(1)119898 minus 119861(1)119898119898 cos 119898ℎ [sin 119898ℎ minus sin 119898 (ℎ minus 119889)]+ 2119886119862(1)0 (119889 minus 119904) + 119872sum
119898=1
2119864(1)119898119903119898 [sin119898120587minus sin 119903119898 (ℎ minus 119889)]
(58)
22 = int119886minus119886
120601(2)1198772 10038161003816100381610038161003816119911=minus119909 119889119909 = 2119886119864(2)0+ 119872sum119898=1
2119862(2)119898119903119898 tanh (119903119898119886) cos 119903119898 (ℎ minus 119904) + 119886 (ℎ minus 119904)minus 11988633 (ℎ minus 119904)
(59)
33 = int0minus119889
( 120601(3)1198771 10038161003816100381610038161003816119909=minus119886 minus 120601(3)1198773 10038161003816100381610038161003816119909=119886) (119911 minus 120585119911) 119889119911+ intminus119904minus119889
(120601(3)1198772 10038161003816100381610038161003816119909=minus119886 minus 120601(3)1198772 10038161003816100381610038161003816119909=119886) (119911 minus 120585119911) 119889119911minus int119886minus119886
120601(3)1198772 10038161003816100381610038161003816119909=minus119904 119909 119889119909 = 119872sum119898=0
(119860(3)119898 minus 119861(3)119898 )
sdot [119889 sin 119898 (ℎ minus 119889) minus 120585119911 sin 119898 (ℎ minus 119889)119898 cos 119898ℎ
+ cos 119898ℎ minus cos 119898 (ℎ minus 119889)1198982 cos 119898ℎ ] + 119886119862(3)0 (1198892 minus 1199042)
minus 119872sum119898=1
2119864(3)119898119903119898 [119889 sin 119898 (ℎ minus 119889)+ cos 119903119898 (ℎ minus 119904) minus cos 119903119898 (ℎ minus 119889)119903119898 ] + 2119886119862(3)0 120585119911 (119889minus 119904) minus 119872sum
119898=1
2119864(3)119898 120585119911119903119898 sin 119903119898 (ℎ minus 119889) minus 23 119862(3)0 1198863 minus 119872sum119898=1
120576sdot cosh 120574119898 (ℎ minus 119904) 1 minus cos 21198861205741198981205741198982 + 119872sum
119898=1
(120576 cos119898120587minus 120576) [ 119889 sinh 120574119898 (ℎ minus 119889) minus 119904 sinh 120574119898 (ℎ minus 119904)120574119898minus cosh 120574119898 (ℎ minus 119904) minus cosh 120574119898 (ℎ minus 119889)1205741198982 ]minus 119872sum119898=1
119864(3)119898 cos 119903119898 (h minus s)119903119898 sinh 119903119898119886 (2119886 cosh 119903119898119886minus 2 sinh 119903119898119886119903119898 )
(60)
31 = 13 = int0minus119889
( 120601(3)1198771 10038161003816100381610038161003816119909=minus119886 minus 120601(3)1198773 10038161003816100381610038161003816119909=119886) 119889119911+ intminus119904minus119889
( 120601(3)1198772 10038161003816100381610038161003816119909=minus119886 minus 120601(3)1198772 10038161003816100381610038161003816119909=119886) 119889119911 = 119872sum119898=0
(119860(3)119898minus 119861(3)119898 ) sin 119898ℎ minus sin 119898 (ℎ minus 119889)
119898 cos 119898ℎ minus 2 (ℎ minus 119904)sdot [ 11988636 (ℎ minus 119904) + 119886119862(3)0 ] + 119886 (ℎ minus 119904)3 minus 119886 (ℎ minus 119889)33 (ℎ minus 119904)+ 119872sum119898=1
2119864(3)119898119903119898 sin 119903119898 (ℎ minus 119889)
(61)
33 Equation of Motion and RAOs The derived solutions forthe diffraction and radiation problems could now be inte-grated with the equation of motion of the floating breakwatersystem to analyze the dynamic responses of the structuralsystem We have
minus1205962 [119872 + 119898] minus i120596 [119889] + [119904] [1198830] = [1198650] (62)
where [1198830] = [11988301 11988302 11988303]119879 and are vectors indicatingthe motion amplitudes and the wave exciting forces respec-tively [119898] and [119889] are addedmass and dampingmatrices [119872]is the mass matrix and [119878] denotes the hydrostatic restoringforce matrix they can be expressed as
[119872] = [[[
1198980 0 minus11989801198851198660 1198980 0minus1198980119885119866 0 1]]]
(63)
[119878] = [[[
0 0 00 2120588119892119886 00 0 1198980119892119866119872
]]]
(64)
Here 1198980 is the mass per unit length of the floatingbreakwater and 119911119866 is the z-coordinate of the center of gravityThe mass moment of inertia about y-axis I can be computedby 119868 = 11989801199031198662 in which 119903119866 is the radius of gyration in pitchingoscillation and 119866119872 is the metacentric height of the floatingbreakwater system Once 1198830119895 (j=1 2 3) are determined bysolving (62) the RAOs (Response Amplitude Operators) ofthe breakwater system can be calculated as
119877119860119874119895 = 1198830119895119867119895 (119895 = 1 2 3) (65)
4 Results and Discussion
The validation of the wave diffraction solutions in terms ofreflectiontransmission coefficients and hydrodynamic loadsfor a train of monochromatic wave propagating throughthe proposed floating breakwater system that consists of apartially submerged rectangular body and with or withoutthe attached two porous side walls has been carried outin a previous study (Qiao et al [24]) Here additional
Mathematical Problems in Engineering 9
Black et al(1971)Present approach
G=G=infinhs=as=
0010203040506070809
F D
gH
a
3 60 5 71 42ks
(a) Horizontal force
Black et al(1971)Present approach
G=G=infinhs= as=
0010203040506070809
1
F D
gHa
3 60 5 71 42ks
(b) Vertical force
Black et al(1971)Present approach
G=G=infinhs= as=
3 60 4 51 72ks
0
005
01
015
02
025
F D
gH
a2
(c) Moment
Figure 2 Comparison of the dimensionless wave exciting forces and moment versus 1198960119904 for ℎ119904 = 20 119886119904 = 30 11986601 = 11986602 997888rarr infin (a)dimensionless horizontal exciting force (b) dimensionless vertical exciting force (c) dimensionless exciting moment about the rotationalcenter of (0 0)
comparisons of wave diffraction results with other publishednumerical solutions for the limiting case of neglecting theeffect of two porous walls are again presented For thewave radiation problem the motion induced hydrodynamiccoefficients such as added mass and damping coefficient arealso investigated for their variations with changes of waveparameters As a validation those results for the caseswithoutporous walls are compared with limited numerical solutionspublished in literatures Subsequently the effects of variousdimensionless parameters for example porous propertiesincident wave conditions and the geometry of floating bodyto wave-induced hydrodynamic forces hydrodynamic coeffi-cients and RAOs were also examined In our computationsthe first 40 terms in the infinite series of the diffractionpotentials and radiation potentials are taken the same as [7]namely M=40
As an example showing the results comparisons for thediffraction problem Figure 2 illustrates the variations ofthe dimensionless exciting forces in horizontal and verticaldirections versus dimensionless wavelength 1198960119904 for the caseof ℎ119904 = 20 119886119904 = 10 11986601 = 11986602 997888rarr infin The excitingmoments with reference to the rotational center of (00) are
also presented in Figure 2 Here 11986601 = 11986602 997888rarr infin representsthe case of a floating breakwater system without dual porousside walls The numerical results computed by Black et al [4]are also plotted in Figure 2 The comparisons indicate thatthe exciting forces and moments are well predicted by thepresent analytical model Both the analytical solutions andthe numerical results reveal that the horizontal exciting forcesand the exciting moments show rapidly increasing trendinitially until a peak value is reached and then are followedby a gradual decrease of the values with the further increaseof 1198960119904 However the vertical forces as shown in Figure 2(b)decrease continuously with an increase in 1198960119904 According tothe results given in the previous study by Qiao et al [24] itindicated that the effectiveness of the porous walls is reflectedwith the reduction of the vertical hydrodynamic forces andwhen the breakwater system is rotated with reference to thebottom center of the body the induced moment can also bereduced However due to the added additional length of theporous walls the horizontal forces are slightly greater thanthose without porous walls
Figure 3 presents the variations of the dimensionlessadded mass and damping coefficients in surge only motion
10 Mathematical Problems in Engineering
YHZheng et al(2004)
G=G=
G=G=
G=G=
G=G=infin
0
1
2
3
4
5
6m
2
as
963 100 87521 114kh
(a) Added mass
YH Zheng et al(2004)
G=G=infin
G=G=
G=G=
G=G=
963 100 87521 114kh
005
115
225
335
4
d
2
as
(b) Radiation coefficient
Figure 3 Variation of the dimensionless surge-surge hydrodynamic coefficients ((a) for the added mass and (b) for the radiation damping)versus 1198960ℎ under the conditions of different porous-effect parameters for 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16
YHZheng et al(2004)
G=G=
G=G=
G=G=
G=G=infin
963 100 87521 114kh
0
02
04
06
08
1
12
14
m
2
as
(a) Added mass
YHZheng et al(2004)
G=G=
G=G=
G=G=
G=G=infin
minus04minus02
002040608
11214
d
2
as
963 100 87521 114kh
(b) Radiation damping
Figure 4 Variation of the dimensionless heave-heave hydrodynamic coefficients ((a) for the added mass and (b) for the radiation damping)versus 1198960ℎ under the conditions of different porous-effect parameters for 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16
versus 1198960ℎ for the setting of 119904ℎ = 13 119889ℎ = 12 and 119886ℎ =16under the conditions of various porous-effect parametersWhen 11986601 = 11986602 = infin it represents the case withoutconsidering the effect of dual porous walls The presentanalytical solutions for this limiting case can be adopted tocompare to the limited numerical results available in thestudy of [7] The comparison plot is shown in Figure 4 wherethe nearly identical matches between the present analyticalsolutions and the numerical results of [7] can be observedThe results shown in Figure 4 suggest that the addition ofporous walls tends to enhance the added mass and dampingcoefficients In general with a decrease of porous effectparameter the addedmass and damping coefficients increaseFor the variation of addedmass coefficient with porouswalls
it can be seen that a peak value occurred at a long wave (small1198960ℎ value) condition However for the radiation dampingcoefficient it is shown to increase rapidly to its maximumvalue with an increase of 1198960ℎ then a gradual decrease trendis followed by a further increase of 1198960ℎ
The dimensionless heave-heave added mass (119898222120588119886119904)and radiation damping coefficient (119889222120588120596119886119904) versus 1198960ℎunder various 11986601 and 11986602 values for the case of 119904ℎ =13 119889ℎ = 12 119886ℎ = 16 are presented in Figure 4Under the limiting condition of 11986601 = 11986602 = infin (withoutporous walls) the numerical results produced by the study of[7] are also plotted for comparison Again good agreementsbetween the present analytical solutions and the publishednumerical results can be noticed The results in Figure 4(a)
Mathematical Problems in Engineering 11
YHZheng et al(2004)
963 100 87521 114kh
0
02
04
06
08
1
12
14m
2
as
G=G=infin
G=G=
G=G=
G=G=
(a) Added mass
YH Zhang et al(2004)
G=G=infin
G=G=
G=G=
G=G=
0010203040506070809
d
2
as
963 100 87521 114kh
(b) Radiation damping
Figure 5 Variation of the dimensionless pitch-pitch hydrodynamic coefficients ((a) for the added mass and (b) for the radiation damping)versus 1198960ℎ under the conditions of different porous-effect parameters for 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16 with the rotational center set at (00)
suggest that the variations of added mass for the shorterwavelength waves (large 1198960ℎ values) are less sensitive to thechange of porous parameters The presence of dual porouswalls increases the heave-heave added mass which tends toreach its maximum value at the long wave extreme Alsoa decrease in porous parameters results in an increase inheave induced added mass As shown in Figure 4(b) theheave-heave damping coefficients decrease rapidly when 1198960ℎincreases initially Then a gradual changing trend is followedby a further increase in 1198960ℎ Additionally the presence ofdual porous walls tends to reduce the values of heave-heavedamping coefficients
For the pitching induced added mass and dampingcoefficients the results are plotted against 1198960ℎ in Figure 5for various porous-effect parameters 11986601 and 11986602 and underthe settings of 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16 Whenthe rotational center is set at (00) as shown in Figure 5 thepresent analytical solutions match well with the publishednumerical results [7] for the limiting case of no porouswall (ie 11986601 = 11986602 = infin) It can be observed thatthe pitching oscillation induced added mass and dampingcoefficients for the floating breakwater system with dualporous side walls are greater than those without porous wallsWith decrease of the porous effect parameters the addedmass and damping coefficients increase For the added masscoefficients it is found that the variations trends are morecomplex for waves with relatively smaller 1198960ℎ under theconditions of different porous-effect parameters Generallyin cases of weaker porous-effect (for example 11986601 = 11986602 le 2as shown in Figure 5(a)) the value of added mass coefficienttends to approach 14 under the case considered For thevariations of damping coefficient as shown in Figure 5(b) thevarying trends are similar to those of damping coefficient insurge motion
The results presented in Figure 6 are the variations ofthe dimensionless surge-pitch added mass and damping
coefficients versus 1198960ℎ for varying 11986601 and 11986602 The case isfor 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16 with referenceto the rotational center of (00) Clearly the added massand damping coefficients are shown to increase with anincrease in the porous-effect parameters of both porouswalls A and B while the damping coefficients reach theirabsolute maximum values when 1198960ℎ is within the rangeof 1 and 2 The magnitude of surge-pitch hydrodynamiccoefficients especially the damping is greatly influenced bythe permeability of the dual porous walls compared to thesituation with surge or pitching motion only
The analytical solutions for the Response AmplitudeOperators (RAOs) of the floating breakwater system pro-posed in this study are illustrated in Figure 7 as a functionof dimensionless wavelength 1198960ℎ for various porous-effectparameters with the conditions of 119904ℎ = 03 119889ℎ = 05119886ℎ = 025 119903119866119886 = 09 and 119866119872119886 = 01 where therotational center is located at (00) It is found that the varyingtrends of surge RAOs are similar to those of pitching RAOsalthough the peak values of surge RAOs are smaller thanthose of pitching RAOs When the porous-effect parametersdecrease the main peak values of either surge or pitchingRAOs that occurred are shifted to the wave conditions withshorter wavelength Furthermore for the heave RAO itexperiences a typical damped response for the case of singledegree of freedom system and its maximum values of RAOwould move to the lower frequency wave region when theporous-effect parameters decrease
5 Summary and Conclusions
The hydrodynamic performance investigations includingthe hydrodynamic loads radiation induced added massand damping coefficients and dynamic responses relatedRAOs are critical to the study that is associated with theproblems of wave and structure interactions In this study the
12 Mathematical Problems in Engineering
G=G=infin
G=G=
G=G=
G=G=
963 100 87521 114kh
minus4
minus3
minus2
minus1
0
1
m
2
as
(a) Added mass
G=G=infin
G=G=
G=G=
G=G=
963 100 87521 114kh
minus3
minus25
minus2
minus15
minus1
minus05
0
d
2
as
(b) Radiation damping
Figure 6 Variation of the dimensionless surge-pitch hydrodynamic coefficients ((a) for the added mass and (b) for the radiation damping)versus 1198960ℎ under the conditions of different porous-effect parameters for 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16 with the rotational center set at(0 0)
G=G=
G=G=
G=G=
G=G=infin
4515 3525 3 605 2 5 551 4kh
0123456789
1011
|X|A
(a) Surge RAO
G=G=infin
G=G=
G=G=
G=G=
3 60 4 51 2kh
0
05
1
15
2
25
3
35|X
|A
(b) Heave RAO
G=G=
G=G=
G=G=
G=G=infin
05 3515 325 45 65 5541 20kh
0
5
10
15
|X|A
(c) Pitch RAO
Figure 7 Variation of the dimensionless RAO ((a) for surge RAO (b) for heave RAO and (c) for pitch RAO) versus 1198960ℎ under the conditionsof different porous-effect parameters for 119904ℎ = 03 119889ℎ = 05 119886ℎ = 025 119903119866119886 = 09 119866119872119886 = 01 with rotation center at (00)
Mathematical Problems in Engineering 13
hydrodynamic problems considering a floating breakwatersystem that consists of a rectangular body and two verticalattached porous walls are investigated analytically based onthe two-dimensional potential flow theory The expressionsof the diffraction and radiation based velocity potentials andthe associated unknown coefficients for each subdomain arederived by the MEEM approach Subsequently the excitinghydrodynamic forces added mass and radiation dampingcoefficients in terms of surge heave and pitching motionsand the equation ofmotion for the floating breakwater systemare formulated and calculated for various case scenariosby varying the wave conditions and porous effect param-eters Considering the limiting case without porous wallsgood agreements between the present analytical solutionsand published numerical results are achieved This suggeststhat the present analytical model is demonstrated to be areliable engineering tool to provide reasonable estimationsfor the hydrodynamic performances of a floating breakwatersystem with dual side porous walls It is worthy to noticethat the wave loads on the floating structures especiallythe moments when referenced to the rotational center ofbottom center would be reduced with an increase in theporous-effect parameters of the porous walls Additionallythe existence of the dual porous walls could move the peaksof surge and pitching RAOs to the waves of higher frequencywhen the porous effect parameters decrease Overall it canbe concluded that a properly adopted porous-wall systemwhen attached to a floating body can effectively improveits hydrodynamic performance as a breakwater The analyt-ical approach developed in this study can also be furtherextended to assist the theoretical analysis for other similar butimproved floating breakwater systems
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was supported by ldquoFundamental Research Fundsfor the Central Universitiesrdquo (Grand No 017180224)
References
[1] I-H Cho ldquoTransmission coefficients of a floating rectangularbreakwater with porous side platesrdquo International Journal ofNaval Architecture and Ocean Engineering vol 8 no 1 pp 53ndash65 2016
[2] H Hu K-H Wang and A N Williams ldquoWave motion over abreakwater system of a horizontal plate and a vertical porouswallrdquo Ocean Engineering vol 29 no 4 pp 373ndash386 2001
[3] C C Mei and J L Black ldquoScattering of surface waves byrectangular obstacles in water of finite depthrdquo Journal of FluidMechanics vol 38 no 3 pp 499ndash511 1969
[4] J L Black C C Mei and M C G Bray ldquoRadiation andscattering of water waves by rigid bodiesrdquo Journal of FluidMechanics vol 46 no 1 pp 151ndash164 1971
[5] N Drimer Y Agnon and M Stiassnie ldquoA simplified analyticalmodel for a floating breakwater in water of finite depthrdquoAppliedOcean Research vol 14 no 1 pp 33ndash41 1992
[6] J-F Lee ldquoOn the heave radiation of a rectangular structurerdquoOcean Engineering vol 22 no 1 pp 19ndash34 1995
[7] Y H Zheng Y G You and Y M Shen ldquoOn the radiationand diffraction of water waves by a rectangular buoyrdquo OceanEngineering vol 31 no 8-9 pp 1063ndash1082 2004
[8] E D Christensen H B Bingham A P Skou Friis A K Larsenand K L Jensen ldquoAn experimental and numerical study offloating breakwatersrdquo Coastal Engineering Journal vol 137 pp43ndash58 2018
[9] E Koutandos P Prinos andXGironella ldquoFloating breakwatersunder regular and irregular wave forcing Reflection and trans-mission characteristicsrdquo Journal of Hydraulic Research vol 43no 2 pp 174ndash188 2005
[10] G H Dong Y N Zheng Y C Li B Teng C T Guan and D FLin ldquoExperiments on wave transmission coefficients of floatingbreakwatersrdquo Ocean Engineering vol 35 no 8-9 pp 931ndash9382008
[11] A T Chwang ldquoA porous-wavemaker theoryrdquo Journal of FluidMechanics vol 132 pp 395ndash406 1983
[12] Y C Li Y Liu and B Teng ldquoPorous effect parameter of thinpermeable platesrdquo Coastal Engineering Journal vol 48 no 4pp 309ndash336 2006
[13] S R Manam and M Sivanesan ldquoScattering of water wavesby vertical porous barriers an analytical approachrdquo WaveMotion An International Journal Reporting Research on WavePhenomena vol 67 pp 89ndash101 2016
[14] A T Chwang and Z Dong ldquoWave-trapping due to a porousplaterdquo in Proceedings of 15 ONR symposium on Naval Hydrody-namics pp 407ndash414 1985
[15] S Zhu and A T Chwang ldquoAnalytical study of porous waveabsorberrdquo Journal of Engineering Mechanics vol 127 no 4 pp326ndash332 2001
[16] H Behera and T Sahoo ldquoHydroelastic analysis of gravity waveinteraction with submerged horizontal flexible porous platerdquoJournal of Fluids and Structures vol 54 pp 643ndash660 2015
[17] T L Yip and A T Chwang ldquoPerforated wall breakwater withinternal horizontal platerdquo Journal of Engineering Mechanics vol126 no 5 pp 533ndash538 2000
[18] Y Liu and H-J Li ldquoWave scattering by dual submergedhorizontal porous plates Further resultsrdquo Ocean Engineeringvol 81 no 5 pp 158ndash163 2014
[19] I H Cho and M H Kim ldquoWave absorbing system usinginclined perforated platesrdquo Journal of Fluid Mechanics vol 608pp 1ndash20 2008
[20] K-H Wang and X Ren ldquoWater waves on flexible and porousbreakwatersrdquo Journal of Engineering Mechanics vol 119 no 5pp 1025ndash1047 1993
[21] K H Wang and X Ren ldquoWave interaction with a concentricporous cylinder systemrdquo Ocean Engineering vol 21 no 4 pp343ndash360 1994
[22] F Zhao T Zhang R Wan L Huang X Wang and WBao ldquoHydrodynamic loads acting on a circular porous platehorizontally submerged in wavesrdquo Ocean Engineering vol 136pp 168ndash177 2017
14 Mathematical Problems in Engineering
[23] A Mondal and R Gayen ldquoWave interaction with dual circularporous platesrdquo Journal of Marine Science and Application vol14 no 4 pp 366ndash375 2015
[24] W L Qiao K-H Wang and Y Q Sun ldquoScattering of waterwaves by a floating body with two vertically attached porouswallsrdquo Journal of Engineering Mechanics vol 144 no 2 ArticleID 04017162 2018
[25] S W Twu and D T Lin ldquoOn a highly effective wave absorberrdquoCoastal Engineering Journal vol 15 no 4 pp 389ndash405 1991
[26] X P Yu ldquoDiffraction of water waves by porous breakwatersrdquoJournal of Waterway Port Coastal and Ocean Engineering vol121 no 6 pp 275ndash282 1995
[27] I J Losada M A Losada and A Baquerizo ldquoAn analyticalmethod to evaluate the efficiency of porous screens as wavedampersrdquo Applied Ocean Research vol 15 no 4 pp 207ndash2151993
[28] G Besio and M A Losada ldquoSediment transport patterns atTrafalgar offshore windfarmrdquo Ocean Engineering vol 35 no 7pp 653ndash665 2008
[29] H Song and L Tao ldquoShort-crestedwave interactionwith a con-centric porous cylindrical structurerdquo Applied Ocean Researchvol 29 no 4 pp 199ndash209 2007
[30] L Xiao Y Kou L Tao and L Yang ldquoComparative studyof hydrodynamic performances of breakwaters with double-layered perforated walls attached to ring-shaped very largefloating structuresrdquo Ocean Engineering vol 111 pp 279ndash2912016
[31] T L Yip and A T Chwang ldquoWater wave control by submergedpitching porous platerdquo Journal of Engineering Mechanics vol124 no 4 pp 428ndash434 1998
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2 Mathematical Problems in Engineering
performances of rectangular boxes of fixed single and fixeddouble subject to the heave motion were estimated under theactions of either regular or irregular water waves [9] Laterthe wave absorbing effectiveness of different types of floatingbreakwaters such as single box double boxes or board netwas also investigated [10]
Due to the advantage of energy dissipation the water-wave absorbing phenomena have been observed when wavespass through a porousmedium that is a porous structure caneffectively attenuate both reflected and transmitted wavesFollowing the porous wavemaker theory of Chwang [11] con-siderable studies related to the interaction between porousstructures and water waves were performed theoretically andexperimentally especially the porous barriers or the porousstructures attached to the floating breakwaters The waveenergy dissipation of thin permeable wall was verified byconducting sets of experiments [12] Analytical approacheshave been carried out by Manam and Sivanesan [13] andChwang and Dong [14] as well as Zhu and Chwang [15] tostudy the reflection and transmission problem of water wavespropagating past vertical porous barriers The hydroelasticissues between gravity waves and a submerged horizontalflexible porous plate were investigated by Behera and Sahoo[16] The studies of perforated wall breakwaters and dualsubmerged horizontal porous plates were given in [17 18]respectively In order to examine the effectiveness of waveabsorption by porous plates the interactions between dif-ferent types of porous plates and oblique monochromaticwaves were investigated by Cho and Kim [19] The conceptof porous barrier was extended by Wang and Ren [20 21]to analytically study the interaction between water wavesand a flexible porous breakwater or in a three-dimensionaldomain a concentric porous cylinder systemThe diffractionproblems of either a circular porous plate submerged hori-zontally or a system with dual circular porous plates wereinvestigated based on the linear potential flow theory [2223] Recently the scattering of water waves by a rectangularfloating body that attached with two vertically porous wallswas investigated analytically and experimentally by Qiao etal [24] In practices the types of porous structure are variousdepending on the requirements of engineering problemhowever single or several vertical porous plates (includingbarrier and circular) and porous cylinder are the mostcommon in engineering fieldThe vertical porous plates wereoften employed to absorb waves to protect coastal [25 26]furthermore a set of porous plates were placed at the bottomof a wave tank to absorb waves [27]The porous cylinder wasemployed to protect structure in Trafalgar offshore windfarm[28] almost for the same purpose a concentric two-cylindersystem involving an interior impermeable cylinder andexternal porous cylinder was designed for Ekofisk gravityoffshore structure in the North Sea [29] then the conceptof single-layered porous cylinder was extended to double-layered perforated cylinder by Xiao et al [30] and theexperimentswere conducted to investigate the hydrodynamicperformances
To achieve the combination of the advantages of a floatingbody and porous wall features a new breakwater systemthat consists of a partially submerged pontoon-like body
and two attached porous side walls is proposed in thisstudy to provide a protected region sheltering from waveaction The results of wave reflectiontransmission and waveinduced hydrodynamic loads under the condition that thebreakwater system is in a fixed position were given in Qiaoet al [24] The investigation of wave interactions with thebreakwater system is further extended to include the effectof the induced motions of the floating breakwater systemthe so-called radiation problem Under the assumptions oflinear incident waves and small translational and rotationalmotions of surge heave and pitch this study focuses on theinvestigation of hydrodynamic coefficients such as the addedmass and damping coefficient and the dynamic responsesof the floating breakwater system through developing ananalytical model with solutions of diffracted and radiatedwaves and the equation of motion which is formulated andsolved The solutions of the velocity potentials in the definedregions including the diffraction and radiation velocitypotentials are obtained by solving the formulated governingequations as well as the boundary and matching conditionsTo verify the analytical solutions derived from this studysome limited numerical results published in literatures wereadopted for comparisons Additionally the effects of theincident wave parameters and porous walls conditions on thehydrodynamic coefficients and RAOs (Response AmplitudeOperators) are examined and discussed
2 Theoretical Formulations and Solutions
The problem statement as illustrated in Figure 1 showsa train of right-going small-amplitude Airy waves withthe frequency 120596 wave height H and wave length 120582 tointeract normally with a floating breakwater system thatconsists of a partially submerged rectangular structure andtwo attached fully submerged side porous walls Subjectto the wave induced hydrodynamic loads the breakwatersystem responds with two translational motions and onerotational motion ie surge heave and pitching motionsThe Cartesian coordinates system is employed to definethis two-dimensional boundary value problem The x-axisis along the still water surface where 119909 = 0 is set at thecenter of the floating breakwater system and the z-axis pointsvertically upwards with an origin located at the still watersurface The rectangular structure with a length of 2a anda submergence of s is initially positioned in a domain withundisturbed constant water depth h Both of the thin porouswalls have a length of d-s
In this study the fluid is assumed to be incompressibleand inviscid and the motion irrational Subsequently thepotential flow theory providing that the velocity potentialsatisfies the Laplace Equation with the defined boundaryconditions could be applied to solve thewaves and breakwaterinteraction problem As shown in Figure 1 the entire fluiddomain is divided into three subdomains namely regions III and III respectively Region I is the subdomain upstreamof the breakwater system (119909 le 119886 minusℎ le 119911 le 0) region II is thesubdomain between the two porous walls (minus119886 le 119909 le 119886 minus119889 le119911 le minus119904) and region III is the subdomain downstream of thebreakwater system (119886 le 119909 minusℎ le 119911 le 0) In the fluid domain
Mathematical Problems in Engineering 3
X
Z
sd
h
Porous wall
2a
surface
bottomI II III
Figure 1 Schematic diagram of a floating body with attached two porous walls under the action of an incident wave and forced surge heaveand pitching motions
the Laplace equation of a complete velocity potential servesas the basic equation to describe the fluid motion namely
nabla2120601 = 0 (1)
The fluid velocity potential 120601 includes three componentsincident wave potential 120601119868 diffracted wave potential 120601119863 andthe radiated wave potential Φ119877 which covers the inducedmotions of the floating breakwater system In this studythe dynamic motions of surge heave and pitching areconsidered Thus the velocity potential 120601 can be expressedas
120601 = 120601119868 + 120601119863 + 3sum119895=1
Φ(119871)119877 (2)
where the subscript L = 1 2 3 donates the surge heaveand pitching motions of the breakwater system respectivelyIn this study the incident wave potential is given as
120601119868 = minusi1198921198672120596 119890i1198960(119909+119886) cosh 1198960 (119911 + ℎ)cosh 1198960ℎ 119890minusi120596119905 (3)
where 1198960 is the wave number120596 is the wave frequency and119892 is the gravitational acceleration Here 1198960 and 120596 satisfy theusual dispersion relation
1205962 = 1198921198960 tanh (1198960ℎ) (4)
21 eoretical Formulations and Solutions of the Incidentand Diffracted Velocity Potentials To solve the wave radi-ation problem in a linear wave system the wave inducedhydrodynamic forces and moment according to the wavediffraction approach by assuming the body is in a fixedposition are required to be determined priorly The wavediffraction solutions for the proposed breakwater systemwithtwo attached porous walls have been obtained in a previousstudy given inQiao et al [24] and Yip and Chwang [31] Hereonly limited formulations and solutions are summarizedbelow Let us define the wave diffraction potentials as 120601119863119894119894 = 1 2 3 stands for the subdomains I II and III respectivelyThe Laplace equations of 120601119863119894 can be solved with the uses ofthe linearized kinematic and dynamic free surface boundarycondition bottom boundary condition no flow condition at
the body surface andmatching conditions at the interfaces ofporous wall locations including the porous flow conditionsexpressed as (Chwang [11])
1205971206011198631120597119909 = 1205971206011198632120597119909 = 1198871120583 (1199011198631 minus 1199011198632)119886119905 119909 = minus119886 minus119889 le 119911 le minus119904
(5)
1205971206011198633120597119909 = 1205971206011198632120597119909 = 1198872120583 (1199011198632 minus 1199011198633)119886119905 119909 = minus119886 minus119889 le 119911 le minus119904
(6)
where 1198871 and 1198872 having the dimension of length arerespectively material constants for the porous walls A andB 119901119863119894 (i=1 2 3) are the dynamic pressures induced bythe incident and diffracted velocity potentials and 120583 is thedynamic viscosity of the fluid The wave diffraction solutionsare derived as
1206011198631 = infinsum119898=0
[119868119898119890119896119898(119909+119886) + 119865119898119890minus119896119898(119909+119886)] cos 119898 (119911 + ℎ)cos 119898ℎ
sdot 119890minusi120596119905(7)
1206011198632 = (1198620119909 + 1198640) 119890minusi120596119905+ infinsum119898=1
[119862119898 cosh (119903119898119909)cosh (119903119898119886) + 119864119898 sinh (119903119898119909)
sinh (119903119898119886) ]sdot cos 119903119898 (119911 + ℎ) 119890minusi120596119905
(8)
1206011198633 = infinsumm=0
[119879119898119890119896119898(119909minus119886)] cos 119898 (119911 + ℎ)cos 119898ℎ 119890minusi120596119905 (9)
where 119865119898 119862119898 119864119898 and 119879119898 (119898 = 0 1 2 3 ) areunknown complex coefficients to be determined 119868119898 119898 and119903119898 (119898 = 0 1 2 3 ) are expressed separately as
119868119898 = minusi1198921198672120596 0 0 0 (10)
119898 = i1198960 minus1198961 minus1198962 minus1198963 minus119896119898 (11)
119903119898 = 119898120587ℎ minus 119904 (12)
4 Mathematical Problems in Engineering
Here 119896119898 can be calculated from
1205962 = minus119892119896119898 tanh (119896119898ℎ) (13)
Substituting (7) (8) and (9) into the following derivedmatching equations
int0minusℎ
1205971206011198631120597119909 cos 119899 (119911 + ℎ) 119889119911= intminus119904minusℎ
1205971206011198632120597119909 cos 119899 (119911 + ℎ) 119889119911119909 = minus119886 (119899 = 0 1 2 3 )
(14)
int0minusℎ
1205971206011198633120597119909 cos 119899 (119911 + ℎ) 119889119911= intminus119904minusℎ
1205971206011198632120597119909 cos 119899 (119911 + ℎ) 119889119911119909 = minus119886 (119899 = 0 1 2 3 )
(15)
intminus119889minusℎ
1206011198631 cos 119903119899 (119911 + ℎ) 119889119911 + intminus119904minus119889
1198631 cos 119903119899 (119911 + ℎ) 119889119911= intminus119904minusℎ
1206011198632 cos 119903119899 (119911 + ℎ) 119889119911119909 = minus119886 (119899 = 0 1 2 )
(16)
intminus119889minusℎ
1206011198633 cos 119903119899 (119911 + ℎ) 119889119911 + intminus119904minus119889
1198633 cos 119903119899 (119911 + ℎ) 119889119911= intminus119904minusℎ
1206011198632 cos 119903119899 (119911 + ℎ) 119889119911119909 = minus119886 (119899 = 0 1 2 )
(17)
and utilizing the orthogonality properties of solutioneigenfunctions the unknown coefficients 119862119899 119863119899 119877119899 119879119899 (119899 =0 1 2 3 119872) for the diffraction potentials can be deter-mined by solving the 4(M+1) system of algebraic equationsHere the terms with summation of infinite series are approx-imated by a large but finite number of M terms In (16) and(17) 1198631 and 1198633 are given as
1198631 = infinsum119898=0
[(119868119898 + 119865119898) minus 119898i119866011198960 (119868119898 minus 119865119898)]
sdot cos 119898 (119911 + ℎ)cos 119898ℎ
(18)
1198633 = infinsum119898=0
(1 + 119898i119866011198960) 119879119898 cos 119898 (119911 + ℎ)
cos 119898ℎ (19)
where 11986601 = 1205881205961198871(1205831198960) and 11986602 = 1205881205961198872(1205831198960) aredimensionless parameters defined according to Chwang [11]to represent the porous effect parameters of porous walls Aand B respectively
22 eoretical Formulations of the Radiated Velocity Poten-tials For the radiation problem the displacements of thefloating breakwater system in motion are given as
119883119895 = 1198830119895119890minusi120596119905 119895 = 1 2 3 (20)
where 119883119900119895 (119895 = 1 2 3) are the amplitudes of surge heaveand pitching motions respectively Let 120601(119871)119877119894 (119871 = 1 2 3 119894 =1 2 3) be the spatial radiated potentials describing the small-amplitude oscillations where 119871119905ℎ indices denote respectivelythe induced surge heave and pitching motions while the 119894119905ℎindices represent the subdomains of I II and III respectivelyThen the total radiated potentials in any of the flow domainscan be expressed as
Φ(119871)119877 = 119883119895120601(119871)119877 = Re [minusi1205961198830119895120601(119871)119877 (119909 119911) 119890minusi120596119905] (21)
As the velocity potentials120601(119871)119877 satisfy the Laplace equationwe have
nabla2120601(119871)119877119894 = 0 (119871 = 1 2 3 119894 = 1 2 3) (22)
The boundary conditions including the free surfaceboundary conditions bottom boundary conditions and thebody surface conditions for the surge heave and pitchingoscillations are listed as follows respectively
120597120601(1)119877119894120597119911 = 0 (119894 = 1 2 3) 119911 = ℎ1205972120601(1)1198771198941205971199052 + 120597120601(1)119877119894120597119911 = 0 (119894 = 1 3) 119911 = 0 |119909| ge 119886
120597120601(1)1198772120597119911 = 0 119911 = minus119904 |119909| le 119886(23)
120597120601(2)119877119894120597119911 = 0 (119894 = 1 2 3) 119911 = ℎ1205972120601(2)1198771198941205971199052 + 119892 120597120601(2)119877119894120597119911 = 0 (119894 = 1 3) 119911 = 0 |119909| ge 119886
120597120601(2)1198772120597119911 = 1 119911 = minus119904 |119909| le 119886(24)
120597120601(3)119877119894120597119911 = 0 (119894 = 1 2 3) 119911 = ℎ1205972120601(3)1198771198941205971199052 + 119892 120597120601(1)119877119894120597119911 = 0 (119894 = 1 3) 119911 = 0 |119909| ge 119886
120597120601(3)119877119894120597119899 = i120596 [(119911 minus 120585119911) 119899119909 minus (119909 minus 120585119909) 119899119911]119900119899 119887119900119889119910 119904119906119903119891119886119888119890
(25)
where 119899 = (119899119909 119899119911) is the unit normal vector and (120585119909 120585119911)is the rotational center of the pitching oscillation With theassumption that the rotational center is along the z axis wecan get 120585119909 = 0
Mathematical Problems in Engineering 5
23 Analytical Solutions of the Radiated Velocity PotentialsSimilar to the derivation of the solutions of the diffractedvelocity potentials the application of the method of sepa-ration variables to the governing equation (22) under theboundary conditions (23) (24) and (25) yields the radiatedvelocity potentials as follows
120601(119871)1198771 = infinsum119898=0
119860(119871)119898 119890minus119896119898(119909+119886) cos 119898 (119911 + ℎ)cos 119898ℎ 119890minusi120596119905 (26)
120601(1)1198772 = (119862(1)0 119909 + 119864(1)0 ) 119890minusi120596119905+ infinsum119898=1
[119862(1)119898 cosh (119903119898119909)cosh (119903119886119909) + 119864(1)119898 sinh (119903119898119909)
sinh (119903119886119909) ]sdot cos 119903119898 (119911 + ℎ) 119890minusi120596119905
(27)
120601(2)1198772 = (119911 + ℎ)2 minus 11990922 (ℎ minus 119904) + (119862(2)0 119909 + 119864(2)0 ) 119890minusi120596119905
+ infinsum119898=1
[119862(2)119898 cosh (119903119898119909)cosh (119903119886119909) + 119864(2)119898 sinh (119903119898119909)
sinh (119903119886119909) ]sdot cos 119903119898 (119911 + ℎ) 119890minusi120596119905
(28)
120601(3)1198772 = infinsum119898=1
120576 cos [120574119898 (119909 minus 119886)] cosh [120574119898 (119911 + ℎ)]+ (119862(3)0 119909 + 119864(3)0 ) 119890minusi120596119905+ infinsum119898=1
[119862(3)119898 cosh (119903119898119909)cosh (119903119886119909) + 119864(3)119898 sinh (119903119898119909)
sinh (119903119886119909) ]sdot cos 119903119898 (119911 + ℎ) 119890minusi120596119905
(29)
120601(119871)1198773 = infinsum119898=0
119861(119871)119898 119890119896119898(119909minus119886) cos 119898 (119911 + ℎ)cos 119898ℎ 119890minusi120596119905 (30)
where
120574119898 = 1198981205872119886 (119898 = 1 2 3 ) (31)
Combining the floating body surface condition at 119911 = minus119904minus119886 le 119909 le 119886 and the orthogonality of cos[120574119898(119909 minus 119886)] over theintegral interval of minus119886 le 119909 le 119886 the expression of 120576 can bederived as
120576 = cos 2119886120574119898sinh [120574119898 (ℎ minus 119904)] 1198861205741198983 (32)
24 Determination of theUnknownCoefficientsatAppearedin the Solutions of Radiated Velocity Potentials The solutionsof the unknown coefficients that appeared in the radiationpotentials can be again obtained by utilizing the matchingconditions at 119909 = plusmn119886 under the principle of continuouspressure and normal velocity We have the following
(a) Matching Conditions for the Surge-Induced Potential
120601(1)1198772 = 120601(1)1198771 minusℎ le 119911 le minus119889 119909 = minus119886120601(1)1198773 minusℎ le 119911 le minus119889 119909 = 119886 (33a)
120597120601(1)1198771120597119909 = 120597120601(1)1198772120597119909 minusℎ le 119911 le minus119889 119909 = minus1198861 minus119904 le 119911 le 0 119909 = 119886 (33b)
120597120601(1)1198773120597119909 = 120597120601(1)1198772120597119909 minusℎ le 119911 le minus119889 119909 = minus1198861 minus119904 le 119911 le 0 119909 = 119886 (33c)
(b) Matching Conditions for the Heave-Induced Potential
120601(2)1198772 = 120601(2)1198771 minusℎ le 119911 le minus119889 119909 = minus119886120601(2)1198773 minusℎ le 119911 le minus119889 119909 = 119886 (34a)
120597120601(2)1198771120597119909 = 120597120601(2)1198772120597119909 minusℎ le 119911 le minus119889 119909 = minus1198861 minus119904 le 119911 le 0 119909 = 119886 (34b)
120597120601(2)1198773120597119909 = 120597120601(2)1198772120597119909 minusℎ le 119911 le minus119889 119909 = minus1198861 minus119904 le 119911 le 0 119909 = 119886 (34c)
(c) Matching Conditions for the Pitching-Induced Potential
120601(3)1198772 = 120601(3)1198771 minusℎ le 119911 le minus119889 119909 = minus119886120601(3)1198773 minusℎ le 119911 le minus119889 119909 = 119886 (35a)
120597120601(3)1198771120597119909 = 120597120601(3)1198772120597119909 minusℎ le 119911 le minus119889 119909 = minus119886119911 minus 120585119911 minus119904 le 119911 le 0 119909 = 119886 (35b)
120597120601(3)1198773120597119909 = 120597120601(3)1198772120597119909 minusℎ le 119911 le minus119889 119909 = minus119886119911 minus 120585119911 minus119904 le 119911 le 0 119909 = 119886 (35c)
For the porous walls A and B at the boundaries 119909 =plusmn119886 the condition of continuity of normal velocity for flowspassing through the porous walls A and B leads to
120597120601(119871)1198771120597119909 = 120597120601(119871)1198772120597119909 = 119882119860 (119911 119905)119871 = (1 2 3) minus ℎ le 119911 le minus119889 119909 = minus119886
(36a)
120597120601(119871)1198773120597119909 = 120597120601(119871)1198772120597119909 = 119882119861 (119911 119905)119871 = (1 2 3) minus ℎ le 119911 le minus119889 119909 = minus119886
(36b)
where the porous flow velocities 119882119860(119911 119905) and 119882119861(119911 119905) areassumed to obey Darcys law that is they are linearly pro-portional to the pressure difference induced by the radiated
6 Mathematical Problems in Engineering
motions across their corresponding porous wall (Chwang[11]) With the application of the linearized Bernoulli equa-tion we have
119882119860 (119911 119905) = minus11986601 [ 1198960120596 (120597120601(119871)1198771120597119905 minus 120597120601(119871)1198772120597119905 )](119871 = 1 2 3) 119909 = minus119886 minus119889 le 119911 le minus119904
(37a)
119882119861 (119911 119905) = minus11986602 [ 1198960120596 (120597120601(119871)1198772120597119905 minus 120597120601(119871)1198773120597119905 )](119871 = 1 2 3) 119909 = 119886 minus119889 le 119911 le minus119904
(37b)
Utilizing the orthogonality property of eigenfunctionscos 119899(119911 + ℎ) for 120601(119871)1198771 and 120601(119871)1198773 (119871 = 1 2 3) over the integralinterval of minusℎ le 119911 le 0 the following equations can beformulated by the matching conditions of (33b) (33c) (34b)(34c) (35b) and (35c) as
int0minusℎ
120597120601(1)1198771120597119909 cos 119899 (119911 + ℎ) 119889119911= intminus119904minusℎ
120597120601(1)1198772120597119909 cos 119899 (119911 + ℎ) 119889119911+ int0minus119904
1 cos 119899 (119911 + ℎ) 119889119911 (119909 = minus119886)(38)
int0minusℎ
120597120601(1)1198773120597119909 cos 119899 (119911 + ℎ) 119889119911= intminus119904minusℎ
120597120601(1)1198772120597119909 cos 119899 (119911 + ℎ) 119889119911+ int0minus119904
1 cos 119899 (119911 + ℎ) 119889119911 (119909 = 119886)(39)
int0minusℎ
120597120601(2)1198771120597119909 cos 119899 (119911 + ℎ) 119889119911= intminus119904minusℎ
120597120601(2)1198772120597119909 cos 119899 (119911 + ℎ) 119889119911 (119909 = minus119886)(40)
int0minusℎ
120597120601(2)1198773120597119909 cos 119899 (119911 + ℎ) 119889119911= intminus119904minusℎ
120597120601(2)1198772120597119909 cos 119899 (119911 + ℎ) 119889119911 (119909 = 119886)(41)
int0minusℎ
120597120601(3)1198771120597119909 cos 119899 (119911 + ℎ) 119889119911= intminus119904minusℎ
120597120601(3)1198772120597119909 cos 119899 (119911 + ℎ) 119889119911+ int0minus119904
(119911 minus 120585119911) cos 119899 (119911 + ℎ) 119889119911 (119909 = minus119886)(42)
int0minusℎ
120597120601(3)1198773120597119909 cos 119899 (119911 + ℎ) 119889119911
= intminus119904minusℎ
120597120601(3)1198772120597119909 cos 119899 (119911 + ℎ) 119889119911+ int0minus119904
(119911 minus 120585119911) cos 119899 (119911 + ℎ) 119889119911 (119909 = 119886)
(43)
Additionally by combining (36a) (36b) (37a) and (37b)with the matching conditions given in (33a) (34a) and(35a) respectively and applying the orthogonality propertyof eigenfunctions cos 119903119899(119911 + ℎ) for 120601(119871)1198772 (L = 1 2 3) over theintegral interval of minusℎ le 119911 le minus119904 we have
intminus119904minusℎ
120601(119871)1198772 cos 119903119899 (119911 + ℎ) 119889119911= intminus119889minusℎ
120601(119871)1198771 cos 119903119899 (119911 + ℎ) 119889119911+ intminus119904minus119889
(119871)1198771 cos 119903119899 (119911 + ℎ) 119889119911(119909 = minus119886 119871 = 1 2 3)
(44)
intminus119904minusℎ
120601(119871)1198772 cos 119903119899 (119911 + ℎ) 119889119911= intminus119889minusℎ
120601(119871)1198773 cos 119903119899 (119911 + ℎ) 119889119911+ intminus119904minus119889
(119871)1198773 cos 119903119899 (119911 + ℎ) 119889119911(119909 = 119886 119871 = 1 2 3)
(45)
where
(1)1198771 = infinsum119898=0
119860(1)119898 (1 + 119898i119866011198960)
cos 119898 (119911 + ℎ)cos 119898ℎ
+ 1i119866011198960
(46)
(2)1198771 = infinsum119898=0
119860(2)119898 (1 + 119898i119866011198960)
cos 119898 (119911 + ℎ)cos 119898ℎ (47)
(3)1198771 = infinsum119898=0
119860(3)119898 (1 + 119898i119866011198960)
cos 119898 (119911 + ℎ)cos 119898ℎ
+ 119911i119866011198960
(48)
Mathematical Problems in Engineering 7
(1)1198773 = infinsum119898=0
119861(1)119898 (1 + 119898i119866021198960)
cos 119898 (119911 + ℎ)cos 119898ℎ
minus 1i119866021198960
(49)
(2)1198773 = infinsum119898=0
119861(2)119898 (1 minus 119898i119866021198960)
cos 119898 (119911 + ℎ)cos 119898ℎ (50)
(3)1198773 = infinsum119898=0
119861(3)119898 (1 + 119898i119866021198960)
cos 119898 (119911 + ℎ)cos 119898ℎ
minus 119911i119866021198960
(51)
Again the summation of terms with the infinite seriesis approximated by a large but finite numbers of termseg a total of M terms Then the unknown coefficients119860(1)119898 119862(1)119898 119864(1)119898 119861(1)119898 (119898 = 0 1 2 119872) [4(M+1) unknownstotally] can be determined by solving the 4(M+1) sys-tem of algebraic equations formulated in (38) (39) (44)and (45) (for L=1) Similarly the unknown coefficients119860(2)119898 119862(2)119898 119864(2)119898 119861(2)119898 (119898 = 0 1 2 119872) can be calculatedfrom the 4(M+1) system of algebraic equations expressedin (40) (41) (44) and (45) (for L=2) while the coefficients119860(3)119898 119862(3)119898 119864(3)119898 119861(3)119898 (119898 = 0 1 2 119872) can be evaluated bysolving the 4(M+1) system of algebraic equations formulatedin (42) (43) (44) and (45) (for L=3) After the determinationof the above unknown coefficients a series of engineeringproperties important to practical applications could be cal-culated for example the hydrodynamic coefficients and themotion induced dynamic responses of the breakwater system
3 Exciting Forces andHydrodynamic Coefficients
31 Exciting Forces To determine the hydrodynamic coeffi-cients related to the inducedmotions of the floating breakwa-ter system the exciting forces and moment of the combinedincident and diffracted waves on the breakwater need tobe computed as inputs for solving the equation of motionBy integrating the dynamic pressures with the use of thevelocity potentials of incident and diffracted waves (see (7)-(9)) over the wet surface the hydrodynamic forces alongthe x direction (1198651198631) and along the z direction (1198651198632) can beformulated respectively as (Qiao et al [24])
1198651198631= i120588120596 infinsum
119898=0
(119868119898 + 119865119898 minus 119879119898) [sin 119898ℎ minus sin 119898 (ℎ minus 119889)]119898 cos 119898ℎ
sdot 119890minusi120596119905 + i120588120596 lowast 2119886 (119889 minus 119904) 1198620119890minusi120596119905minus i120588120596 infinsum
119898=0
2119864119898119903119898 sin 119903119898 (ℎ minus 119889) 119890minusi120596119905(52)
1198651198632 = i120596120588 [21198861198640 + 2 infinsum119898=1
119862119898119903119898 tanh (119903119898119886) cos 119903119898 (ℎ minus 119904)]sdot 119890minusi120596119905
(53)
Also the wave exciting moment (1198651198633) that is referencedto a rotational center (0120585119911) can be derived as
1198651198633 = i120588120596
(119868119898 + 119865119898 minus 119879119898)
sdot [[[[[[
119889 sin 119898 (ℎ minus 119889) minus 120585119911 sin 119898ℎ + 120585119911 sin 119898 (ℎ minus 119889)119898 cos 119898ℎ
+cos 119898ℎ minus cos 119898 (ℎ minus 119889)1198982 cos 119898ℎ
]]]]]]
sdot 119890minusi120596119905 + i120588120596 1198861198620 (1198892 minus 1199042)
minus infinsum119898=1
2119864119898119903119898 [119889 sin 119903119898 (ℎ minus 119889)
+ cos 119903119898 (ℎ minus 119904) minus cos 119903119898 (ℎ minus 119889)119903119898 ] 119890minusi120596119905
+ i120588120596 [21198861198620120585119911 (119889 minus 119904) minus infinsum119898=1
2119864119898120585119911119903119898 sin 119903119898 (ℎ minus 119889)]
sdot 119890minusi120596119905 minus i120588120596 2119862011988633 + infinsum119898=1
[ 2119886119864119898 cos 119903119898 (ℎ minus 119904)119903119898 tanh (119903119898119886)minus 2119864119898 cos 119903119898 (ℎ minus 119904)1199031198982 ] 119890minusi120596119905
(54)
32 Hydrodynamic Coefficients The radiation forces inducedby the translational and rotational motions of the floatingbreakwater system can be calculated from the derived radi-ation potentials utilizing the linearized Bernoulli equation as
119865119871119896 = i120588120596 intΓ
Φ(119871)119877119896 119890minusi120596119905119899119895119889Γ= 12058812059621198830119895 int
Γ120601(119871)119877119896 119890minusi120596119905119899119895119889Γ
= 12059621198830119895119898119871119896119890minusi120596119905 + i1205961198830119895119889119871119896119890minusi120596119905(55)
where 119898119871119896 is the radiation-induced added mass and the119889119871119896 denotes the radiation damping The expressions arerespectively
119898119871119896 = Re(120588 intΓ
120601(119871)119877119896119899119895119889Γ) = Re (120588119871119896) (56)
119889119871119896 = Im (120588120596 intΓ
120601(119871)119877119896119899119895119889Γ) = Im (120588120596119871119896) (57)
8 Mathematical Problems in Engineering
119871119896 can be computed with formulations given below
11 = (int0minus119889
120601(1)1198771 10038161003816100381610038161003816119909=minus119886 119889119911 minus int0minus119889
120601(1)1198773 10038161003816100381610038161003816119909=119886 119889119911)+ (intminus119904minus119889
120601(1)1198772 10038161003816100381610038161003816119909=minus119886 119889119911 minus intminus119904minus119889
120601(1)1198772 10038161003816100381610038161003816119909=119886 119889119911)= 119872sum119898=1
119860(1)119898 minus 119861(1)119898119898 cos 119898ℎ [sin 119898ℎ minus sin 119898 (ℎ minus 119889)]+ 2119886119862(1)0 (119889 minus 119904) + 119872sum
119898=1
2119864(1)119898119903119898 [sin119898120587minus sin 119903119898 (ℎ minus 119889)]
(58)
22 = int119886minus119886
120601(2)1198772 10038161003816100381610038161003816119911=minus119909 119889119909 = 2119886119864(2)0+ 119872sum119898=1
2119862(2)119898119903119898 tanh (119903119898119886) cos 119903119898 (ℎ minus 119904) + 119886 (ℎ minus 119904)minus 11988633 (ℎ minus 119904)
(59)
33 = int0minus119889
( 120601(3)1198771 10038161003816100381610038161003816119909=minus119886 minus 120601(3)1198773 10038161003816100381610038161003816119909=119886) (119911 minus 120585119911) 119889119911+ intminus119904minus119889
(120601(3)1198772 10038161003816100381610038161003816119909=minus119886 minus 120601(3)1198772 10038161003816100381610038161003816119909=119886) (119911 minus 120585119911) 119889119911minus int119886minus119886
120601(3)1198772 10038161003816100381610038161003816119909=minus119904 119909 119889119909 = 119872sum119898=0
(119860(3)119898 minus 119861(3)119898 )
sdot [119889 sin 119898 (ℎ minus 119889) minus 120585119911 sin 119898 (ℎ minus 119889)119898 cos 119898ℎ
+ cos 119898ℎ minus cos 119898 (ℎ minus 119889)1198982 cos 119898ℎ ] + 119886119862(3)0 (1198892 minus 1199042)
minus 119872sum119898=1
2119864(3)119898119903119898 [119889 sin 119898 (ℎ minus 119889)+ cos 119903119898 (ℎ minus 119904) minus cos 119903119898 (ℎ minus 119889)119903119898 ] + 2119886119862(3)0 120585119911 (119889minus 119904) minus 119872sum
119898=1
2119864(3)119898 120585119911119903119898 sin 119903119898 (ℎ minus 119889) minus 23 119862(3)0 1198863 minus 119872sum119898=1
120576sdot cosh 120574119898 (ℎ minus 119904) 1 minus cos 21198861205741198981205741198982 + 119872sum
119898=1
(120576 cos119898120587minus 120576) [ 119889 sinh 120574119898 (ℎ minus 119889) minus 119904 sinh 120574119898 (ℎ minus 119904)120574119898minus cosh 120574119898 (ℎ minus 119904) minus cosh 120574119898 (ℎ minus 119889)1205741198982 ]minus 119872sum119898=1
119864(3)119898 cos 119903119898 (h minus s)119903119898 sinh 119903119898119886 (2119886 cosh 119903119898119886minus 2 sinh 119903119898119886119903119898 )
(60)
31 = 13 = int0minus119889
( 120601(3)1198771 10038161003816100381610038161003816119909=minus119886 minus 120601(3)1198773 10038161003816100381610038161003816119909=119886) 119889119911+ intminus119904minus119889
( 120601(3)1198772 10038161003816100381610038161003816119909=minus119886 minus 120601(3)1198772 10038161003816100381610038161003816119909=119886) 119889119911 = 119872sum119898=0
(119860(3)119898minus 119861(3)119898 ) sin 119898ℎ minus sin 119898 (ℎ minus 119889)
119898 cos 119898ℎ minus 2 (ℎ minus 119904)sdot [ 11988636 (ℎ minus 119904) + 119886119862(3)0 ] + 119886 (ℎ minus 119904)3 minus 119886 (ℎ minus 119889)33 (ℎ minus 119904)+ 119872sum119898=1
2119864(3)119898119903119898 sin 119903119898 (ℎ minus 119889)
(61)
33 Equation of Motion and RAOs The derived solutions forthe diffraction and radiation problems could now be inte-grated with the equation of motion of the floating breakwatersystem to analyze the dynamic responses of the structuralsystem We have
minus1205962 [119872 + 119898] minus i120596 [119889] + [119904] [1198830] = [1198650] (62)
where [1198830] = [11988301 11988302 11988303]119879 and are vectors indicatingthe motion amplitudes and the wave exciting forces respec-tively [119898] and [119889] are addedmass and dampingmatrices [119872]is the mass matrix and [119878] denotes the hydrostatic restoringforce matrix they can be expressed as
[119872] = [[[
1198980 0 minus11989801198851198660 1198980 0minus1198980119885119866 0 1]]]
(63)
[119878] = [[[
0 0 00 2120588119892119886 00 0 1198980119892119866119872
]]]
(64)
Here 1198980 is the mass per unit length of the floatingbreakwater and 119911119866 is the z-coordinate of the center of gravityThe mass moment of inertia about y-axis I can be computedby 119868 = 11989801199031198662 in which 119903119866 is the radius of gyration in pitchingoscillation and 119866119872 is the metacentric height of the floatingbreakwater system Once 1198830119895 (j=1 2 3) are determined bysolving (62) the RAOs (Response Amplitude Operators) ofthe breakwater system can be calculated as
119877119860119874119895 = 1198830119895119867119895 (119895 = 1 2 3) (65)
4 Results and Discussion
The validation of the wave diffraction solutions in terms ofreflectiontransmission coefficients and hydrodynamic loadsfor a train of monochromatic wave propagating throughthe proposed floating breakwater system that consists of apartially submerged rectangular body and with or withoutthe attached two porous side walls has been carried outin a previous study (Qiao et al [24]) Here additional
Mathematical Problems in Engineering 9
Black et al(1971)Present approach
G=G=infinhs=as=
0010203040506070809
F D
gH
a
3 60 5 71 42ks
(a) Horizontal force
Black et al(1971)Present approach
G=G=infinhs= as=
0010203040506070809
1
F D
gHa
3 60 5 71 42ks
(b) Vertical force
Black et al(1971)Present approach
G=G=infinhs= as=
3 60 4 51 72ks
0
005
01
015
02
025
F D
gH
a2
(c) Moment
Figure 2 Comparison of the dimensionless wave exciting forces and moment versus 1198960119904 for ℎ119904 = 20 119886119904 = 30 11986601 = 11986602 997888rarr infin (a)dimensionless horizontal exciting force (b) dimensionless vertical exciting force (c) dimensionless exciting moment about the rotationalcenter of (0 0)
comparisons of wave diffraction results with other publishednumerical solutions for the limiting case of neglecting theeffect of two porous walls are again presented For thewave radiation problem the motion induced hydrodynamiccoefficients such as added mass and damping coefficient arealso investigated for their variations with changes of waveparameters As a validation those results for the caseswithoutporous walls are compared with limited numerical solutionspublished in literatures Subsequently the effects of variousdimensionless parameters for example porous propertiesincident wave conditions and the geometry of floating bodyto wave-induced hydrodynamic forces hydrodynamic coeffi-cients and RAOs were also examined In our computationsthe first 40 terms in the infinite series of the diffractionpotentials and radiation potentials are taken the same as [7]namely M=40
As an example showing the results comparisons for thediffraction problem Figure 2 illustrates the variations ofthe dimensionless exciting forces in horizontal and verticaldirections versus dimensionless wavelength 1198960119904 for the caseof ℎ119904 = 20 119886119904 = 10 11986601 = 11986602 997888rarr infin The excitingmoments with reference to the rotational center of (00) are
also presented in Figure 2 Here 11986601 = 11986602 997888rarr infin representsthe case of a floating breakwater system without dual porousside walls The numerical results computed by Black et al [4]are also plotted in Figure 2 The comparisons indicate thatthe exciting forces and moments are well predicted by thepresent analytical model Both the analytical solutions andthe numerical results reveal that the horizontal exciting forcesand the exciting moments show rapidly increasing trendinitially until a peak value is reached and then are followedby a gradual decrease of the values with the further increaseof 1198960119904 However the vertical forces as shown in Figure 2(b)decrease continuously with an increase in 1198960119904 According tothe results given in the previous study by Qiao et al [24] itindicated that the effectiveness of the porous walls is reflectedwith the reduction of the vertical hydrodynamic forces andwhen the breakwater system is rotated with reference to thebottom center of the body the induced moment can also bereduced However due to the added additional length of theporous walls the horizontal forces are slightly greater thanthose without porous walls
Figure 3 presents the variations of the dimensionlessadded mass and damping coefficients in surge only motion
10 Mathematical Problems in Engineering
YHZheng et al(2004)
G=G=
G=G=
G=G=
G=G=infin
0
1
2
3
4
5
6m
2
as
963 100 87521 114kh
(a) Added mass
YH Zheng et al(2004)
G=G=infin
G=G=
G=G=
G=G=
963 100 87521 114kh
005
115
225
335
4
d
2
as
(b) Radiation coefficient
Figure 3 Variation of the dimensionless surge-surge hydrodynamic coefficients ((a) for the added mass and (b) for the radiation damping)versus 1198960ℎ under the conditions of different porous-effect parameters for 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16
YHZheng et al(2004)
G=G=
G=G=
G=G=
G=G=infin
963 100 87521 114kh
0
02
04
06
08
1
12
14
m
2
as
(a) Added mass
YHZheng et al(2004)
G=G=
G=G=
G=G=
G=G=infin
minus04minus02
002040608
11214
d
2
as
963 100 87521 114kh
(b) Radiation damping
Figure 4 Variation of the dimensionless heave-heave hydrodynamic coefficients ((a) for the added mass and (b) for the radiation damping)versus 1198960ℎ under the conditions of different porous-effect parameters for 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16
versus 1198960ℎ for the setting of 119904ℎ = 13 119889ℎ = 12 and 119886ℎ =16under the conditions of various porous-effect parametersWhen 11986601 = 11986602 = infin it represents the case withoutconsidering the effect of dual porous walls The presentanalytical solutions for this limiting case can be adopted tocompare to the limited numerical results available in thestudy of [7] The comparison plot is shown in Figure 4 wherethe nearly identical matches between the present analyticalsolutions and the numerical results of [7] can be observedThe results shown in Figure 4 suggest that the addition ofporous walls tends to enhance the added mass and dampingcoefficients In general with a decrease of porous effectparameter the addedmass and damping coefficients increaseFor the variation of addedmass coefficient with porouswalls
it can be seen that a peak value occurred at a long wave (small1198960ℎ value) condition However for the radiation dampingcoefficient it is shown to increase rapidly to its maximumvalue with an increase of 1198960ℎ then a gradual decrease trendis followed by a further increase of 1198960ℎ
The dimensionless heave-heave added mass (119898222120588119886119904)and radiation damping coefficient (119889222120588120596119886119904) versus 1198960ℎunder various 11986601 and 11986602 values for the case of 119904ℎ =13 119889ℎ = 12 119886ℎ = 16 are presented in Figure 4Under the limiting condition of 11986601 = 11986602 = infin (withoutporous walls) the numerical results produced by the study of[7] are also plotted for comparison Again good agreementsbetween the present analytical solutions and the publishednumerical results can be noticed The results in Figure 4(a)
Mathematical Problems in Engineering 11
YHZheng et al(2004)
963 100 87521 114kh
0
02
04
06
08
1
12
14m
2
as
G=G=infin
G=G=
G=G=
G=G=
(a) Added mass
YH Zhang et al(2004)
G=G=infin
G=G=
G=G=
G=G=
0010203040506070809
d
2
as
963 100 87521 114kh
(b) Radiation damping
Figure 5 Variation of the dimensionless pitch-pitch hydrodynamic coefficients ((a) for the added mass and (b) for the radiation damping)versus 1198960ℎ under the conditions of different porous-effect parameters for 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16 with the rotational center set at (00)
suggest that the variations of added mass for the shorterwavelength waves (large 1198960ℎ values) are less sensitive to thechange of porous parameters The presence of dual porouswalls increases the heave-heave added mass which tends toreach its maximum value at the long wave extreme Alsoa decrease in porous parameters results in an increase inheave induced added mass As shown in Figure 4(b) theheave-heave damping coefficients decrease rapidly when 1198960ℎincreases initially Then a gradual changing trend is followedby a further increase in 1198960ℎ Additionally the presence ofdual porous walls tends to reduce the values of heave-heavedamping coefficients
For the pitching induced added mass and dampingcoefficients the results are plotted against 1198960ℎ in Figure 5for various porous-effect parameters 11986601 and 11986602 and underthe settings of 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16 Whenthe rotational center is set at (00) as shown in Figure 5 thepresent analytical solutions match well with the publishednumerical results [7] for the limiting case of no porouswall (ie 11986601 = 11986602 = infin) It can be observed thatthe pitching oscillation induced added mass and dampingcoefficients for the floating breakwater system with dualporous side walls are greater than those without porous wallsWith decrease of the porous effect parameters the addedmass and damping coefficients increase For the added masscoefficients it is found that the variations trends are morecomplex for waves with relatively smaller 1198960ℎ under theconditions of different porous-effect parameters Generallyin cases of weaker porous-effect (for example 11986601 = 11986602 le 2as shown in Figure 5(a)) the value of added mass coefficienttends to approach 14 under the case considered For thevariations of damping coefficient as shown in Figure 5(b) thevarying trends are similar to those of damping coefficient insurge motion
The results presented in Figure 6 are the variations ofthe dimensionless surge-pitch added mass and damping
coefficients versus 1198960ℎ for varying 11986601 and 11986602 The case isfor 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16 with referenceto the rotational center of (00) Clearly the added massand damping coefficients are shown to increase with anincrease in the porous-effect parameters of both porouswalls A and B while the damping coefficients reach theirabsolute maximum values when 1198960ℎ is within the rangeof 1 and 2 The magnitude of surge-pitch hydrodynamiccoefficients especially the damping is greatly influenced bythe permeability of the dual porous walls compared to thesituation with surge or pitching motion only
The analytical solutions for the Response AmplitudeOperators (RAOs) of the floating breakwater system pro-posed in this study are illustrated in Figure 7 as a functionof dimensionless wavelength 1198960ℎ for various porous-effectparameters with the conditions of 119904ℎ = 03 119889ℎ = 05119886ℎ = 025 119903119866119886 = 09 and 119866119872119886 = 01 where therotational center is located at (00) It is found that the varyingtrends of surge RAOs are similar to those of pitching RAOsalthough the peak values of surge RAOs are smaller thanthose of pitching RAOs When the porous-effect parametersdecrease the main peak values of either surge or pitchingRAOs that occurred are shifted to the wave conditions withshorter wavelength Furthermore for the heave RAO itexperiences a typical damped response for the case of singledegree of freedom system and its maximum values of RAOwould move to the lower frequency wave region when theporous-effect parameters decrease
5 Summary and Conclusions
The hydrodynamic performance investigations includingthe hydrodynamic loads radiation induced added massand damping coefficients and dynamic responses relatedRAOs are critical to the study that is associated with theproblems of wave and structure interactions In this study the
12 Mathematical Problems in Engineering
G=G=infin
G=G=
G=G=
G=G=
963 100 87521 114kh
minus4
minus3
minus2
minus1
0
1
m
2
as
(a) Added mass
G=G=infin
G=G=
G=G=
G=G=
963 100 87521 114kh
minus3
minus25
minus2
minus15
minus1
minus05
0
d
2
as
(b) Radiation damping
Figure 6 Variation of the dimensionless surge-pitch hydrodynamic coefficients ((a) for the added mass and (b) for the radiation damping)versus 1198960ℎ under the conditions of different porous-effect parameters for 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16 with the rotational center set at(0 0)
G=G=
G=G=
G=G=
G=G=infin
4515 3525 3 605 2 5 551 4kh
0123456789
1011
|X|A
(a) Surge RAO
G=G=infin
G=G=
G=G=
G=G=
3 60 4 51 2kh
0
05
1
15
2
25
3
35|X
|A
(b) Heave RAO
G=G=
G=G=
G=G=
G=G=infin
05 3515 325 45 65 5541 20kh
0
5
10
15
|X|A
(c) Pitch RAO
Figure 7 Variation of the dimensionless RAO ((a) for surge RAO (b) for heave RAO and (c) for pitch RAO) versus 1198960ℎ under the conditionsof different porous-effect parameters for 119904ℎ = 03 119889ℎ = 05 119886ℎ = 025 119903119866119886 = 09 119866119872119886 = 01 with rotation center at (00)
Mathematical Problems in Engineering 13
hydrodynamic problems considering a floating breakwatersystem that consists of a rectangular body and two verticalattached porous walls are investigated analytically based onthe two-dimensional potential flow theory The expressionsof the diffraction and radiation based velocity potentials andthe associated unknown coefficients for each subdomain arederived by the MEEM approach Subsequently the excitinghydrodynamic forces added mass and radiation dampingcoefficients in terms of surge heave and pitching motionsand the equation ofmotion for the floating breakwater systemare formulated and calculated for various case scenariosby varying the wave conditions and porous effect param-eters Considering the limiting case without porous wallsgood agreements between the present analytical solutionsand published numerical results are achieved This suggeststhat the present analytical model is demonstrated to be areliable engineering tool to provide reasonable estimationsfor the hydrodynamic performances of a floating breakwatersystem with dual side porous walls It is worthy to noticethat the wave loads on the floating structures especiallythe moments when referenced to the rotational center ofbottom center would be reduced with an increase in theporous-effect parameters of the porous walls Additionallythe existence of the dual porous walls could move the peaksof surge and pitching RAOs to the waves of higher frequencywhen the porous effect parameters decrease Overall it canbe concluded that a properly adopted porous-wall systemwhen attached to a floating body can effectively improveits hydrodynamic performance as a breakwater The analyt-ical approach developed in this study can also be furtherextended to assist the theoretical analysis for other similar butimproved floating breakwater systems
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was supported by ldquoFundamental Research Fundsfor the Central Universitiesrdquo (Grand No 017180224)
References
[1] I-H Cho ldquoTransmission coefficients of a floating rectangularbreakwater with porous side platesrdquo International Journal ofNaval Architecture and Ocean Engineering vol 8 no 1 pp 53ndash65 2016
[2] H Hu K-H Wang and A N Williams ldquoWave motion over abreakwater system of a horizontal plate and a vertical porouswallrdquo Ocean Engineering vol 29 no 4 pp 373ndash386 2001
[3] C C Mei and J L Black ldquoScattering of surface waves byrectangular obstacles in water of finite depthrdquo Journal of FluidMechanics vol 38 no 3 pp 499ndash511 1969
[4] J L Black C C Mei and M C G Bray ldquoRadiation andscattering of water waves by rigid bodiesrdquo Journal of FluidMechanics vol 46 no 1 pp 151ndash164 1971
[5] N Drimer Y Agnon and M Stiassnie ldquoA simplified analyticalmodel for a floating breakwater in water of finite depthrdquoAppliedOcean Research vol 14 no 1 pp 33ndash41 1992
[6] J-F Lee ldquoOn the heave radiation of a rectangular structurerdquoOcean Engineering vol 22 no 1 pp 19ndash34 1995
[7] Y H Zheng Y G You and Y M Shen ldquoOn the radiationand diffraction of water waves by a rectangular buoyrdquo OceanEngineering vol 31 no 8-9 pp 1063ndash1082 2004
[8] E D Christensen H B Bingham A P Skou Friis A K Larsenand K L Jensen ldquoAn experimental and numerical study offloating breakwatersrdquo Coastal Engineering Journal vol 137 pp43ndash58 2018
[9] E Koutandos P Prinos andXGironella ldquoFloating breakwatersunder regular and irregular wave forcing Reflection and trans-mission characteristicsrdquo Journal of Hydraulic Research vol 43no 2 pp 174ndash188 2005
[10] G H Dong Y N Zheng Y C Li B Teng C T Guan and D FLin ldquoExperiments on wave transmission coefficients of floatingbreakwatersrdquo Ocean Engineering vol 35 no 8-9 pp 931ndash9382008
[11] A T Chwang ldquoA porous-wavemaker theoryrdquo Journal of FluidMechanics vol 132 pp 395ndash406 1983
[12] Y C Li Y Liu and B Teng ldquoPorous effect parameter of thinpermeable platesrdquo Coastal Engineering Journal vol 48 no 4pp 309ndash336 2006
[13] S R Manam and M Sivanesan ldquoScattering of water wavesby vertical porous barriers an analytical approachrdquo WaveMotion An International Journal Reporting Research on WavePhenomena vol 67 pp 89ndash101 2016
[14] A T Chwang and Z Dong ldquoWave-trapping due to a porousplaterdquo in Proceedings of 15 ONR symposium on Naval Hydrody-namics pp 407ndash414 1985
[15] S Zhu and A T Chwang ldquoAnalytical study of porous waveabsorberrdquo Journal of Engineering Mechanics vol 127 no 4 pp326ndash332 2001
[16] H Behera and T Sahoo ldquoHydroelastic analysis of gravity waveinteraction with submerged horizontal flexible porous platerdquoJournal of Fluids and Structures vol 54 pp 643ndash660 2015
[17] T L Yip and A T Chwang ldquoPerforated wall breakwater withinternal horizontal platerdquo Journal of Engineering Mechanics vol126 no 5 pp 533ndash538 2000
[18] Y Liu and H-J Li ldquoWave scattering by dual submergedhorizontal porous plates Further resultsrdquo Ocean Engineeringvol 81 no 5 pp 158ndash163 2014
[19] I H Cho and M H Kim ldquoWave absorbing system usinginclined perforated platesrdquo Journal of Fluid Mechanics vol 608pp 1ndash20 2008
[20] K-H Wang and X Ren ldquoWater waves on flexible and porousbreakwatersrdquo Journal of Engineering Mechanics vol 119 no 5pp 1025ndash1047 1993
[21] K H Wang and X Ren ldquoWave interaction with a concentricporous cylinder systemrdquo Ocean Engineering vol 21 no 4 pp343ndash360 1994
[22] F Zhao T Zhang R Wan L Huang X Wang and WBao ldquoHydrodynamic loads acting on a circular porous platehorizontally submerged in wavesrdquo Ocean Engineering vol 136pp 168ndash177 2017
14 Mathematical Problems in Engineering
[23] A Mondal and R Gayen ldquoWave interaction with dual circularporous platesrdquo Journal of Marine Science and Application vol14 no 4 pp 366ndash375 2015
[24] W L Qiao K-H Wang and Y Q Sun ldquoScattering of waterwaves by a floating body with two vertically attached porouswallsrdquo Journal of Engineering Mechanics vol 144 no 2 ArticleID 04017162 2018
[25] S W Twu and D T Lin ldquoOn a highly effective wave absorberrdquoCoastal Engineering Journal vol 15 no 4 pp 389ndash405 1991
[26] X P Yu ldquoDiffraction of water waves by porous breakwatersrdquoJournal of Waterway Port Coastal and Ocean Engineering vol121 no 6 pp 275ndash282 1995
[27] I J Losada M A Losada and A Baquerizo ldquoAn analyticalmethod to evaluate the efficiency of porous screens as wavedampersrdquo Applied Ocean Research vol 15 no 4 pp 207ndash2151993
[28] G Besio and M A Losada ldquoSediment transport patterns atTrafalgar offshore windfarmrdquo Ocean Engineering vol 35 no 7pp 653ndash665 2008
[29] H Song and L Tao ldquoShort-crestedwave interactionwith a con-centric porous cylindrical structurerdquo Applied Ocean Researchvol 29 no 4 pp 199ndash209 2007
[30] L Xiao Y Kou L Tao and L Yang ldquoComparative studyof hydrodynamic performances of breakwaters with double-layered perforated walls attached to ring-shaped very largefloating structuresrdquo Ocean Engineering vol 111 pp 279ndash2912016
[31] T L Yip and A T Chwang ldquoWater wave control by submergedpitching porous platerdquo Journal of Engineering Mechanics vol124 no 4 pp 428ndash434 1998
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Mathematical Problems in Engineering 3
X
Z
sd
h
Porous wall
2a
surface
bottomI II III
Figure 1 Schematic diagram of a floating body with attached two porous walls under the action of an incident wave and forced surge heaveand pitching motions
the Laplace equation of a complete velocity potential servesas the basic equation to describe the fluid motion namely
nabla2120601 = 0 (1)
The fluid velocity potential 120601 includes three componentsincident wave potential 120601119868 diffracted wave potential 120601119863 andthe radiated wave potential Φ119877 which covers the inducedmotions of the floating breakwater system In this studythe dynamic motions of surge heave and pitching areconsidered Thus the velocity potential 120601 can be expressedas
120601 = 120601119868 + 120601119863 + 3sum119895=1
Φ(119871)119877 (2)
where the subscript L = 1 2 3 donates the surge heaveand pitching motions of the breakwater system respectivelyIn this study the incident wave potential is given as
120601119868 = minusi1198921198672120596 119890i1198960(119909+119886) cosh 1198960 (119911 + ℎ)cosh 1198960ℎ 119890minusi120596119905 (3)
where 1198960 is the wave number120596 is the wave frequency and119892 is the gravitational acceleration Here 1198960 and 120596 satisfy theusual dispersion relation
1205962 = 1198921198960 tanh (1198960ℎ) (4)
21 eoretical Formulations and Solutions of the Incidentand Diffracted Velocity Potentials To solve the wave radi-ation problem in a linear wave system the wave inducedhydrodynamic forces and moment according to the wavediffraction approach by assuming the body is in a fixedposition are required to be determined priorly The wavediffraction solutions for the proposed breakwater systemwithtwo attached porous walls have been obtained in a previousstudy given inQiao et al [24] and Yip and Chwang [31] Hereonly limited formulations and solutions are summarizedbelow Let us define the wave diffraction potentials as 120601119863119894119894 = 1 2 3 stands for the subdomains I II and III respectivelyThe Laplace equations of 120601119863119894 can be solved with the uses ofthe linearized kinematic and dynamic free surface boundarycondition bottom boundary condition no flow condition at
the body surface andmatching conditions at the interfaces ofporous wall locations including the porous flow conditionsexpressed as (Chwang [11])
1205971206011198631120597119909 = 1205971206011198632120597119909 = 1198871120583 (1199011198631 minus 1199011198632)119886119905 119909 = minus119886 minus119889 le 119911 le minus119904
(5)
1205971206011198633120597119909 = 1205971206011198632120597119909 = 1198872120583 (1199011198632 minus 1199011198633)119886119905 119909 = minus119886 minus119889 le 119911 le minus119904
(6)
where 1198871 and 1198872 having the dimension of length arerespectively material constants for the porous walls A andB 119901119863119894 (i=1 2 3) are the dynamic pressures induced bythe incident and diffracted velocity potentials and 120583 is thedynamic viscosity of the fluid The wave diffraction solutionsare derived as
1206011198631 = infinsum119898=0
[119868119898119890119896119898(119909+119886) + 119865119898119890minus119896119898(119909+119886)] cos 119898 (119911 + ℎ)cos 119898ℎ
sdot 119890minusi120596119905(7)
1206011198632 = (1198620119909 + 1198640) 119890minusi120596119905+ infinsum119898=1
[119862119898 cosh (119903119898119909)cosh (119903119898119886) + 119864119898 sinh (119903119898119909)
sinh (119903119898119886) ]sdot cos 119903119898 (119911 + ℎ) 119890minusi120596119905
(8)
1206011198633 = infinsumm=0
[119879119898119890119896119898(119909minus119886)] cos 119898 (119911 + ℎ)cos 119898ℎ 119890minusi120596119905 (9)
where 119865119898 119862119898 119864119898 and 119879119898 (119898 = 0 1 2 3 ) areunknown complex coefficients to be determined 119868119898 119898 and119903119898 (119898 = 0 1 2 3 ) are expressed separately as
119868119898 = minusi1198921198672120596 0 0 0 (10)
119898 = i1198960 minus1198961 minus1198962 minus1198963 minus119896119898 (11)
119903119898 = 119898120587ℎ minus 119904 (12)
4 Mathematical Problems in Engineering
Here 119896119898 can be calculated from
1205962 = minus119892119896119898 tanh (119896119898ℎ) (13)
Substituting (7) (8) and (9) into the following derivedmatching equations
int0minusℎ
1205971206011198631120597119909 cos 119899 (119911 + ℎ) 119889119911= intminus119904minusℎ
1205971206011198632120597119909 cos 119899 (119911 + ℎ) 119889119911119909 = minus119886 (119899 = 0 1 2 3 )
(14)
int0minusℎ
1205971206011198633120597119909 cos 119899 (119911 + ℎ) 119889119911= intminus119904minusℎ
1205971206011198632120597119909 cos 119899 (119911 + ℎ) 119889119911119909 = minus119886 (119899 = 0 1 2 3 )
(15)
intminus119889minusℎ
1206011198631 cos 119903119899 (119911 + ℎ) 119889119911 + intminus119904minus119889
1198631 cos 119903119899 (119911 + ℎ) 119889119911= intminus119904minusℎ
1206011198632 cos 119903119899 (119911 + ℎ) 119889119911119909 = minus119886 (119899 = 0 1 2 )
(16)
intminus119889minusℎ
1206011198633 cos 119903119899 (119911 + ℎ) 119889119911 + intminus119904minus119889
1198633 cos 119903119899 (119911 + ℎ) 119889119911= intminus119904minusℎ
1206011198632 cos 119903119899 (119911 + ℎ) 119889119911119909 = minus119886 (119899 = 0 1 2 )
(17)
and utilizing the orthogonality properties of solutioneigenfunctions the unknown coefficients 119862119899 119863119899 119877119899 119879119899 (119899 =0 1 2 3 119872) for the diffraction potentials can be deter-mined by solving the 4(M+1) system of algebraic equationsHere the terms with summation of infinite series are approx-imated by a large but finite number of M terms In (16) and(17) 1198631 and 1198633 are given as
1198631 = infinsum119898=0
[(119868119898 + 119865119898) minus 119898i119866011198960 (119868119898 minus 119865119898)]
sdot cos 119898 (119911 + ℎ)cos 119898ℎ
(18)
1198633 = infinsum119898=0
(1 + 119898i119866011198960) 119879119898 cos 119898 (119911 + ℎ)
cos 119898ℎ (19)
where 11986601 = 1205881205961198871(1205831198960) and 11986602 = 1205881205961198872(1205831198960) aredimensionless parameters defined according to Chwang [11]to represent the porous effect parameters of porous walls Aand B respectively
22 eoretical Formulations of the Radiated Velocity Poten-tials For the radiation problem the displacements of thefloating breakwater system in motion are given as
119883119895 = 1198830119895119890minusi120596119905 119895 = 1 2 3 (20)
where 119883119900119895 (119895 = 1 2 3) are the amplitudes of surge heaveand pitching motions respectively Let 120601(119871)119877119894 (119871 = 1 2 3 119894 =1 2 3) be the spatial radiated potentials describing the small-amplitude oscillations where 119871119905ℎ indices denote respectivelythe induced surge heave and pitching motions while the 119894119905ℎindices represent the subdomains of I II and III respectivelyThen the total radiated potentials in any of the flow domainscan be expressed as
Φ(119871)119877 = 119883119895120601(119871)119877 = Re [minusi1205961198830119895120601(119871)119877 (119909 119911) 119890minusi120596119905] (21)
As the velocity potentials120601(119871)119877 satisfy the Laplace equationwe have
nabla2120601(119871)119877119894 = 0 (119871 = 1 2 3 119894 = 1 2 3) (22)
The boundary conditions including the free surfaceboundary conditions bottom boundary conditions and thebody surface conditions for the surge heave and pitchingoscillations are listed as follows respectively
120597120601(1)119877119894120597119911 = 0 (119894 = 1 2 3) 119911 = ℎ1205972120601(1)1198771198941205971199052 + 120597120601(1)119877119894120597119911 = 0 (119894 = 1 3) 119911 = 0 |119909| ge 119886
120597120601(1)1198772120597119911 = 0 119911 = minus119904 |119909| le 119886(23)
120597120601(2)119877119894120597119911 = 0 (119894 = 1 2 3) 119911 = ℎ1205972120601(2)1198771198941205971199052 + 119892 120597120601(2)119877119894120597119911 = 0 (119894 = 1 3) 119911 = 0 |119909| ge 119886
120597120601(2)1198772120597119911 = 1 119911 = minus119904 |119909| le 119886(24)
120597120601(3)119877119894120597119911 = 0 (119894 = 1 2 3) 119911 = ℎ1205972120601(3)1198771198941205971199052 + 119892 120597120601(1)119877119894120597119911 = 0 (119894 = 1 3) 119911 = 0 |119909| ge 119886
120597120601(3)119877119894120597119899 = i120596 [(119911 minus 120585119911) 119899119909 minus (119909 minus 120585119909) 119899119911]119900119899 119887119900119889119910 119904119906119903119891119886119888119890
(25)
where 119899 = (119899119909 119899119911) is the unit normal vector and (120585119909 120585119911)is the rotational center of the pitching oscillation With theassumption that the rotational center is along the z axis wecan get 120585119909 = 0
Mathematical Problems in Engineering 5
23 Analytical Solutions of the Radiated Velocity PotentialsSimilar to the derivation of the solutions of the diffractedvelocity potentials the application of the method of sepa-ration variables to the governing equation (22) under theboundary conditions (23) (24) and (25) yields the radiatedvelocity potentials as follows
120601(119871)1198771 = infinsum119898=0
119860(119871)119898 119890minus119896119898(119909+119886) cos 119898 (119911 + ℎ)cos 119898ℎ 119890minusi120596119905 (26)
120601(1)1198772 = (119862(1)0 119909 + 119864(1)0 ) 119890minusi120596119905+ infinsum119898=1
[119862(1)119898 cosh (119903119898119909)cosh (119903119886119909) + 119864(1)119898 sinh (119903119898119909)
sinh (119903119886119909) ]sdot cos 119903119898 (119911 + ℎ) 119890minusi120596119905
(27)
120601(2)1198772 = (119911 + ℎ)2 minus 11990922 (ℎ minus 119904) + (119862(2)0 119909 + 119864(2)0 ) 119890minusi120596119905
+ infinsum119898=1
[119862(2)119898 cosh (119903119898119909)cosh (119903119886119909) + 119864(2)119898 sinh (119903119898119909)
sinh (119903119886119909) ]sdot cos 119903119898 (119911 + ℎ) 119890minusi120596119905
(28)
120601(3)1198772 = infinsum119898=1
120576 cos [120574119898 (119909 minus 119886)] cosh [120574119898 (119911 + ℎ)]+ (119862(3)0 119909 + 119864(3)0 ) 119890minusi120596119905+ infinsum119898=1
[119862(3)119898 cosh (119903119898119909)cosh (119903119886119909) + 119864(3)119898 sinh (119903119898119909)
sinh (119903119886119909) ]sdot cos 119903119898 (119911 + ℎ) 119890minusi120596119905
(29)
120601(119871)1198773 = infinsum119898=0
119861(119871)119898 119890119896119898(119909minus119886) cos 119898 (119911 + ℎ)cos 119898ℎ 119890minusi120596119905 (30)
where
120574119898 = 1198981205872119886 (119898 = 1 2 3 ) (31)
Combining the floating body surface condition at 119911 = minus119904minus119886 le 119909 le 119886 and the orthogonality of cos[120574119898(119909 minus 119886)] over theintegral interval of minus119886 le 119909 le 119886 the expression of 120576 can bederived as
120576 = cos 2119886120574119898sinh [120574119898 (ℎ minus 119904)] 1198861205741198983 (32)
24 Determination of theUnknownCoefficientsatAppearedin the Solutions of Radiated Velocity Potentials The solutionsof the unknown coefficients that appeared in the radiationpotentials can be again obtained by utilizing the matchingconditions at 119909 = plusmn119886 under the principle of continuouspressure and normal velocity We have the following
(a) Matching Conditions for the Surge-Induced Potential
120601(1)1198772 = 120601(1)1198771 minusℎ le 119911 le minus119889 119909 = minus119886120601(1)1198773 minusℎ le 119911 le minus119889 119909 = 119886 (33a)
120597120601(1)1198771120597119909 = 120597120601(1)1198772120597119909 minusℎ le 119911 le minus119889 119909 = minus1198861 minus119904 le 119911 le 0 119909 = 119886 (33b)
120597120601(1)1198773120597119909 = 120597120601(1)1198772120597119909 minusℎ le 119911 le minus119889 119909 = minus1198861 minus119904 le 119911 le 0 119909 = 119886 (33c)
(b) Matching Conditions for the Heave-Induced Potential
120601(2)1198772 = 120601(2)1198771 minusℎ le 119911 le minus119889 119909 = minus119886120601(2)1198773 minusℎ le 119911 le minus119889 119909 = 119886 (34a)
120597120601(2)1198771120597119909 = 120597120601(2)1198772120597119909 minusℎ le 119911 le minus119889 119909 = minus1198861 minus119904 le 119911 le 0 119909 = 119886 (34b)
120597120601(2)1198773120597119909 = 120597120601(2)1198772120597119909 minusℎ le 119911 le minus119889 119909 = minus1198861 minus119904 le 119911 le 0 119909 = 119886 (34c)
(c) Matching Conditions for the Pitching-Induced Potential
120601(3)1198772 = 120601(3)1198771 minusℎ le 119911 le minus119889 119909 = minus119886120601(3)1198773 minusℎ le 119911 le minus119889 119909 = 119886 (35a)
120597120601(3)1198771120597119909 = 120597120601(3)1198772120597119909 minusℎ le 119911 le minus119889 119909 = minus119886119911 minus 120585119911 minus119904 le 119911 le 0 119909 = 119886 (35b)
120597120601(3)1198773120597119909 = 120597120601(3)1198772120597119909 minusℎ le 119911 le minus119889 119909 = minus119886119911 minus 120585119911 minus119904 le 119911 le 0 119909 = 119886 (35c)
For the porous walls A and B at the boundaries 119909 =plusmn119886 the condition of continuity of normal velocity for flowspassing through the porous walls A and B leads to
120597120601(119871)1198771120597119909 = 120597120601(119871)1198772120597119909 = 119882119860 (119911 119905)119871 = (1 2 3) minus ℎ le 119911 le minus119889 119909 = minus119886
(36a)
120597120601(119871)1198773120597119909 = 120597120601(119871)1198772120597119909 = 119882119861 (119911 119905)119871 = (1 2 3) minus ℎ le 119911 le minus119889 119909 = minus119886
(36b)
where the porous flow velocities 119882119860(119911 119905) and 119882119861(119911 119905) areassumed to obey Darcys law that is they are linearly pro-portional to the pressure difference induced by the radiated
6 Mathematical Problems in Engineering
motions across their corresponding porous wall (Chwang[11]) With the application of the linearized Bernoulli equa-tion we have
119882119860 (119911 119905) = minus11986601 [ 1198960120596 (120597120601(119871)1198771120597119905 minus 120597120601(119871)1198772120597119905 )](119871 = 1 2 3) 119909 = minus119886 minus119889 le 119911 le minus119904
(37a)
119882119861 (119911 119905) = minus11986602 [ 1198960120596 (120597120601(119871)1198772120597119905 minus 120597120601(119871)1198773120597119905 )](119871 = 1 2 3) 119909 = 119886 minus119889 le 119911 le minus119904
(37b)
Utilizing the orthogonality property of eigenfunctionscos 119899(119911 + ℎ) for 120601(119871)1198771 and 120601(119871)1198773 (119871 = 1 2 3) over the integralinterval of minusℎ le 119911 le 0 the following equations can beformulated by the matching conditions of (33b) (33c) (34b)(34c) (35b) and (35c) as
int0minusℎ
120597120601(1)1198771120597119909 cos 119899 (119911 + ℎ) 119889119911= intminus119904minusℎ
120597120601(1)1198772120597119909 cos 119899 (119911 + ℎ) 119889119911+ int0minus119904
1 cos 119899 (119911 + ℎ) 119889119911 (119909 = minus119886)(38)
int0minusℎ
120597120601(1)1198773120597119909 cos 119899 (119911 + ℎ) 119889119911= intminus119904minusℎ
120597120601(1)1198772120597119909 cos 119899 (119911 + ℎ) 119889119911+ int0minus119904
1 cos 119899 (119911 + ℎ) 119889119911 (119909 = 119886)(39)
int0minusℎ
120597120601(2)1198771120597119909 cos 119899 (119911 + ℎ) 119889119911= intminus119904minusℎ
120597120601(2)1198772120597119909 cos 119899 (119911 + ℎ) 119889119911 (119909 = minus119886)(40)
int0minusℎ
120597120601(2)1198773120597119909 cos 119899 (119911 + ℎ) 119889119911= intminus119904minusℎ
120597120601(2)1198772120597119909 cos 119899 (119911 + ℎ) 119889119911 (119909 = 119886)(41)
int0minusℎ
120597120601(3)1198771120597119909 cos 119899 (119911 + ℎ) 119889119911= intminus119904minusℎ
120597120601(3)1198772120597119909 cos 119899 (119911 + ℎ) 119889119911+ int0minus119904
(119911 minus 120585119911) cos 119899 (119911 + ℎ) 119889119911 (119909 = minus119886)(42)
int0minusℎ
120597120601(3)1198773120597119909 cos 119899 (119911 + ℎ) 119889119911
= intminus119904minusℎ
120597120601(3)1198772120597119909 cos 119899 (119911 + ℎ) 119889119911+ int0minus119904
(119911 minus 120585119911) cos 119899 (119911 + ℎ) 119889119911 (119909 = 119886)
(43)
Additionally by combining (36a) (36b) (37a) and (37b)with the matching conditions given in (33a) (34a) and(35a) respectively and applying the orthogonality propertyof eigenfunctions cos 119903119899(119911 + ℎ) for 120601(119871)1198772 (L = 1 2 3) over theintegral interval of minusℎ le 119911 le minus119904 we have
intminus119904minusℎ
120601(119871)1198772 cos 119903119899 (119911 + ℎ) 119889119911= intminus119889minusℎ
120601(119871)1198771 cos 119903119899 (119911 + ℎ) 119889119911+ intminus119904minus119889
(119871)1198771 cos 119903119899 (119911 + ℎ) 119889119911(119909 = minus119886 119871 = 1 2 3)
(44)
intminus119904minusℎ
120601(119871)1198772 cos 119903119899 (119911 + ℎ) 119889119911= intminus119889minusℎ
120601(119871)1198773 cos 119903119899 (119911 + ℎ) 119889119911+ intminus119904minus119889
(119871)1198773 cos 119903119899 (119911 + ℎ) 119889119911(119909 = 119886 119871 = 1 2 3)
(45)
where
(1)1198771 = infinsum119898=0
119860(1)119898 (1 + 119898i119866011198960)
cos 119898 (119911 + ℎ)cos 119898ℎ
+ 1i119866011198960
(46)
(2)1198771 = infinsum119898=0
119860(2)119898 (1 + 119898i119866011198960)
cos 119898 (119911 + ℎ)cos 119898ℎ (47)
(3)1198771 = infinsum119898=0
119860(3)119898 (1 + 119898i119866011198960)
cos 119898 (119911 + ℎ)cos 119898ℎ
+ 119911i119866011198960
(48)
Mathematical Problems in Engineering 7
(1)1198773 = infinsum119898=0
119861(1)119898 (1 + 119898i119866021198960)
cos 119898 (119911 + ℎ)cos 119898ℎ
minus 1i119866021198960
(49)
(2)1198773 = infinsum119898=0
119861(2)119898 (1 minus 119898i119866021198960)
cos 119898 (119911 + ℎ)cos 119898ℎ (50)
(3)1198773 = infinsum119898=0
119861(3)119898 (1 + 119898i119866021198960)
cos 119898 (119911 + ℎ)cos 119898ℎ
minus 119911i119866021198960
(51)
Again the summation of terms with the infinite seriesis approximated by a large but finite numbers of termseg a total of M terms Then the unknown coefficients119860(1)119898 119862(1)119898 119864(1)119898 119861(1)119898 (119898 = 0 1 2 119872) [4(M+1) unknownstotally] can be determined by solving the 4(M+1) sys-tem of algebraic equations formulated in (38) (39) (44)and (45) (for L=1) Similarly the unknown coefficients119860(2)119898 119862(2)119898 119864(2)119898 119861(2)119898 (119898 = 0 1 2 119872) can be calculatedfrom the 4(M+1) system of algebraic equations expressedin (40) (41) (44) and (45) (for L=2) while the coefficients119860(3)119898 119862(3)119898 119864(3)119898 119861(3)119898 (119898 = 0 1 2 119872) can be evaluated bysolving the 4(M+1) system of algebraic equations formulatedin (42) (43) (44) and (45) (for L=3) After the determinationof the above unknown coefficients a series of engineeringproperties important to practical applications could be cal-culated for example the hydrodynamic coefficients and themotion induced dynamic responses of the breakwater system
3 Exciting Forces andHydrodynamic Coefficients
31 Exciting Forces To determine the hydrodynamic coeffi-cients related to the inducedmotions of the floating breakwa-ter system the exciting forces and moment of the combinedincident and diffracted waves on the breakwater need tobe computed as inputs for solving the equation of motionBy integrating the dynamic pressures with the use of thevelocity potentials of incident and diffracted waves (see (7)-(9)) over the wet surface the hydrodynamic forces alongthe x direction (1198651198631) and along the z direction (1198651198632) can beformulated respectively as (Qiao et al [24])
1198651198631= i120588120596 infinsum
119898=0
(119868119898 + 119865119898 minus 119879119898) [sin 119898ℎ minus sin 119898 (ℎ minus 119889)]119898 cos 119898ℎ
sdot 119890minusi120596119905 + i120588120596 lowast 2119886 (119889 minus 119904) 1198620119890minusi120596119905minus i120588120596 infinsum
119898=0
2119864119898119903119898 sin 119903119898 (ℎ minus 119889) 119890minusi120596119905(52)
1198651198632 = i120596120588 [21198861198640 + 2 infinsum119898=1
119862119898119903119898 tanh (119903119898119886) cos 119903119898 (ℎ minus 119904)]sdot 119890minusi120596119905
(53)
Also the wave exciting moment (1198651198633) that is referencedto a rotational center (0120585119911) can be derived as
1198651198633 = i120588120596
(119868119898 + 119865119898 minus 119879119898)
sdot [[[[[[
119889 sin 119898 (ℎ minus 119889) minus 120585119911 sin 119898ℎ + 120585119911 sin 119898 (ℎ minus 119889)119898 cos 119898ℎ
+cos 119898ℎ minus cos 119898 (ℎ minus 119889)1198982 cos 119898ℎ
]]]]]]
sdot 119890minusi120596119905 + i120588120596 1198861198620 (1198892 minus 1199042)
minus infinsum119898=1
2119864119898119903119898 [119889 sin 119903119898 (ℎ minus 119889)
+ cos 119903119898 (ℎ minus 119904) minus cos 119903119898 (ℎ minus 119889)119903119898 ] 119890minusi120596119905
+ i120588120596 [21198861198620120585119911 (119889 minus 119904) minus infinsum119898=1
2119864119898120585119911119903119898 sin 119903119898 (ℎ minus 119889)]
sdot 119890minusi120596119905 minus i120588120596 2119862011988633 + infinsum119898=1
[ 2119886119864119898 cos 119903119898 (ℎ minus 119904)119903119898 tanh (119903119898119886)minus 2119864119898 cos 119903119898 (ℎ minus 119904)1199031198982 ] 119890minusi120596119905
(54)
32 Hydrodynamic Coefficients The radiation forces inducedby the translational and rotational motions of the floatingbreakwater system can be calculated from the derived radi-ation potentials utilizing the linearized Bernoulli equation as
119865119871119896 = i120588120596 intΓ
Φ(119871)119877119896 119890minusi120596119905119899119895119889Γ= 12058812059621198830119895 int
Γ120601(119871)119877119896 119890minusi120596119905119899119895119889Γ
= 12059621198830119895119898119871119896119890minusi120596119905 + i1205961198830119895119889119871119896119890minusi120596119905(55)
where 119898119871119896 is the radiation-induced added mass and the119889119871119896 denotes the radiation damping The expressions arerespectively
119898119871119896 = Re(120588 intΓ
120601(119871)119877119896119899119895119889Γ) = Re (120588119871119896) (56)
119889119871119896 = Im (120588120596 intΓ
120601(119871)119877119896119899119895119889Γ) = Im (120588120596119871119896) (57)
8 Mathematical Problems in Engineering
119871119896 can be computed with formulations given below
11 = (int0minus119889
120601(1)1198771 10038161003816100381610038161003816119909=minus119886 119889119911 minus int0minus119889
120601(1)1198773 10038161003816100381610038161003816119909=119886 119889119911)+ (intminus119904minus119889
120601(1)1198772 10038161003816100381610038161003816119909=minus119886 119889119911 minus intminus119904minus119889
120601(1)1198772 10038161003816100381610038161003816119909=119886 119889119911)= 119872sum119898=1
119860(1)119898 minus 119861(1)119898119898 cos 119898ℎ [sin 119898ℎ minus sin 119898 (ℎ minus 119889)]+ 2119886119862(1)0 (119889 minus 119904) + 119872sum
119898=1
2119864(1)119898119903119898 [sin119898120587minus sin 119903119898 (ℎ minus 119889)]
(58)
22 = int119886minus119886
120601(2)1198772 10038161003816100381610038161003816119911=minus119909 119889119909 = 2119886119864(2)0+ 119872sum119898=1
2119862(2)119898119903119898 tanh (119903119898119886) cos 119903119898 (ℎ minus 119904) + 119886 (ℎ minus 119904)minus 11988633 (ℎ minus 119904)
(59)
33 = int0minus119889
( 120601(3)1198771 10038161003816100381610038161003816119909=minus119886 minus 120601(3)1198773 10038161003816100381610038161003816119909=119886) (119911 minus 120585119911) 119889119911+ intminus119904minus119889
(120601(3)1198772 10038161003816100381610038161003816119909=minus119886 minus 120601(3)1198772 10038161003816100381610038161003816119909=119886) (119911 minus 120585119911) 119889119911minus int119886minus119886
120601(3)1198772 10038161003816100381610038161003816119909=minus119904 119909 119889119909 = 119872sum119898=0
(119860(3)119898 minus 119861(3)119898 )
sdot [119889 sin 119898 (ℎ minus 119889) minus 120585119911 sin 119898 (ℎ minus 119889)119898 cos 119898ℎ
+ cos 119898ℎ minus cos 119898 (ℎ minus 119889)1198982 cos 119898ℎ ] + 119886119862(3)0 (1198892 minus 1199042)
minus 119872sum119898=1
2119864(3)119898119903119898 [119889 sin 119898 (ℎ minus 119889)+ cos 119903119898 (ℎ minus 119904) minus cos 119903119898 (ℎ minus 119889)119903119898 ] + 2119886119862(3)0 120585119911 (119889minus 119904) minus 119872sum
119898=1
2119864(3)119898 120585119911119903119898 sin 119903119898 (ℎ minus 119889) minus 23 119862(3)0 1198863 minus 119872sum119898=1
120576sdot cosh 120574119898 (ℎ minus 119904) 1 minus cos 21198861205741198981205741198982 + 119872sum
119898=1
(120576 cos119898120587minus 120576) [ 119889 sinh 120574119898 (ℎ minus 119889) minus 119904 sinh 120574119898 (ℎ minus 119904)120574119898minus cosh 120574119898 (ℎ minus 119904) minus cosh 120574119898 (ℎ minus 119889)1205741198982 ]minus 119872sum119898=1
119864(3)119898 cos 119903119898 (h minus s)119903119898 sinh 119903119898119886 (2119886 cosh 119903119898119886minus 2 sinh 119903119898119886119903119898 )
(60)
31 = 13 = int0minus119889
( 120601(3)1198771 10038161003816100381610038161003816119909=minus119886 minus 120601(3)1198773 10038161003816100381610038161003816119909=119886) 119889119911+ intminus119904minus119889
( 120601(3)1198772 10038161003816100381610038161003816119909=minus119886 minus 120601(3)1198772 10038161003816100381610038161003816119909=119886) 119889119911 = 119872sum119898=0
(119860(3)119898minus 119861(3)119898 ) sin 119898ℎ minus sin 119898 (ℎ minus 119889)
119898 cos 119898ℎ minus 2 (ℎ minus 119904)sdot [ 11988636 (ℎ minus 119904) + 119886119862(3)0 ] + 119886 (ℎ minus 119904)3 minus 119886 (ℎ minus 119889)33 (ℎ minus 119904)+ 119872sum119898=1
2119864(3)119898119903119898 sin 119903119898 (ℎ minus 119889)
(61)
33 Equation of Motion and RAOs The derived solutions forthe diffraction and radiation problems could now be inte-grated with the equation of motion of the floating breakwatersystem to analyze the dynamic responses of the structuralsystem We have
minus1205962 [119872 + 119898] minus i120596 [119889] + [119904] [1198830] = [1198650] (62)
where [1198830] = [11988301 11988302 11988303]119879 and are vectors indicatingthe motion amplitudes and the wave exciting forces respec-tively [119898] and [119889] are addedmass and dampingmatrices [119872]is the mass matrix and [119878] denotes the hydrostatic restoringforce matrix they can be expressed as
[119872] = [[[
1198980 0 minus11989801198851198660 1198980 0minus1198980119885119866 0 1]]]
(63)
[119878] = [[[
0 0 00 2120588119892119886 00 0 1198980119892119866119872
]]]
(64)
Here 1198980 is the mass per unit length of the floatingbreakwater and 119911119866 is the z-coordinate of the center of gravityThe mass moment of inertia about y-axis I can be computedby 119868 = 11989801199031198662 in which 119903119866 is the radius of gyration in pitchingoscillation and 119866119872 is the metacentric height of the floatingbreakwater system Once 1198830119895 (j=1 2 3) are determined bysolving (62) the RAOs (Response Amplitude Operators) ofthe breakwater system can be calculated as
119877119860119874119895 = 1198830119895119867119895 (119895 = 1 2 3) (65)
4 Results and Discussion
The validation of the wave diffraction solutions in terms ofreflectiontransmission coefficients and hydrodynamic loadsfor a train of monochromatic wave propagating throughthe proposed floating breakwater system that consists of apartially submerged rectangular body and with or withoutthe attached two porous side walls has been carried outin a previous study (Qiao et al [24]) Here additional
Mathematical Problems in Engineering 9
Black et al(1971)Present approach
G=G=infinhs=as=
0010203040506070809
F D
gH
a
3 60 5 71 42ks
(a) Horizontal force
Black et al(1971)Present approach
G=G=infinhs= as=
0010203040506070809
1
F D
gHa
3 60 5 71 42ks
(b) Vertical force
Black et al(1971)Present approach
G=G=infinhs= as=
3 60 4 51 72ks
0
005
01
015
02
025
F D
gH
a2
(c) Moment
Figure 2 Comparison of the dimensionless wave exciting forces and moment versus 1198960119904 for ℎ119904 = 20 119886119904 = 30 11986601 = 11986602 997888rarr infin (a)dimensionless horizontal exciting force (b) dimensionless vertical exciting force (c) dimensionless exciting moment about the rotationalcenter of (0 0)
comparisons of wave diffraction results with other publishednumerical solutions for the limiting case of neglecting theeffect of two porous walls are again presented For thewave radiation problem the motion induced hydrodynamiccoefficients such as added mass and damping coefficient arealso investigated for their variations with changes of waveparameters As a validation those results for the caseswithoutporous walls are compared with limited numerical solutionspublished in literatures Subsequently the effects of variousdimensionless parameters for example porous propertiesincident wave conditions and the geometry of floating bodyto wave-induced hydrodynamic forces hydrodynamic coeffi-cients and RAOs were also examined In our computationsthe first 40 terms in the infinite series of the diffractionpotentials and radiation potentials are taken the same as [7]namely M=40
As an example showing the results comparisons for thediffraction problem Figure 2 illustrates the variations ofthe dimensionless exciting forces in horizontal and verticaldirections versus dimensionless wavelength 1198960119904 for the caseof ℎ119904 = 20 119886119904 = 10 11986601 = 11986602 997888rarr infin The excitingmoments with reference to the rotational center of (00) are
also presented in Figure 2 Here 11986601 = 11986602 997888rarr infin representsthe case of a floating breakwater system without dual porousside walls The numerical results computed by Black et al [4]are also plotted in Figure 2 The comparisons indicate thatthe exciting forces and moments are well predicted by thepresent analytical model Both the analytical solutions andthe numerical results reveal that the horizontal exciting forcesand the exciting moments show rapidly increasing trendinitially until a peak value is reached and then are followedby a gradual decrease of the values with the further increaseof 1198960119904 However the vertical forces as shown in Figure 2(b)decrease continuously with an increase in 1198960119904 According tothe results given in the previous study by Qiao et al [24] itindicated that the effectiveness of the porous walls is reflectedwith the reduction of the vertical hydrodynamic forces andwhen the breakwater system is rotated with reference to thebottom center of the body the induced moment can also bereduced However due to the added additional length of theporous walls the horizontal forces are slightly greater thanthose without porous walls
Figure 3 presents the variations of the dimensionlessadded mass and damping coefficients in surge only motion
10 Mathematical Problems in Engineering
YHZheng et al(2004)
G=G=
G=G=
G=G=
G=G=infin
0
1
2
3
4
5
6m
2
as
963 100 87521 114kh
(a) Added mass
YH Zheng et al(2004)
G=G=infin
G=G=
G=G=
G=G=
963 100 87521 114kh
005
115
225
335
4
d
2
as
(b) Radiation coefficient
Figure 3 Variation of the dimensionless surge-surge hydrodynamic coefficients ((a) for the added mass and (b) for the radiation damping)versus 1198960ℎ under the conditions of different porous-effect parameters for 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16
YHZheng et al(2004)
G=G=
G=G=
G=G=
G=G=infin
963 100 87521 114kh
0
02
04
06
08
1
12
14
m
2
as
(a) Added mass
YHZheng et al(2004)
G=G=
G=G=
G=G=
G=G=infin
minus04minus02
002040608
11214
d
2
as
963 100 87521 114kh
(b) Radiation damping
Figure 4 Variation of the dimensionless heave-heave hydrodynamic coefficients ((a) for the added mass and (b) for the radiation damping)versus 1198960ℎ under the conditions of different porous-effect parameters for 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16
versus 1198960ℎ for the setting of 119904ℎ = 13 119889ℎ = 12 and 119886ℎ =16under the conditions of various porous-effect parametersWhen 11986601 = 11986602 = infin it represents the case withoutconsidering the effect of dual porous walls The presentanalytical solutions for this limiting case can be adopted tocompare to the limited numerical results available in thestudy of [7] The comparison plot is shown in Figure 4 wherethe nearly identical matches between the present analyticalsolutions and the numerical results of [7] can be observedThe results shown in Figure 4 suggest that the addition ofporous walls tends to enhance the added mass and dampingcoefficients In general with a decrease of porous effectparameter the addedmass and damping coefficients increaseFor the variation of addedmass coefficient with porouswalls
it can be seen that a peak value occurred at a long wave (small1198960ℎ value) condition However for the radiation dampingcoefficient it is shown to increase rapidly to its maximumvalue with an increase of 1198960ℎ then a gradual decrease trendis followed by a further increase of 1198960ℎ
The dimensionless heave-heave added mass (119898222120588119886119904)and radiation damping coefficient (119889222120588120596119886119904) versus 1198960ℎunder various 11986601 and 11986602 values for the case of 119904ℎ =13 119889ℎ = 12 119886ℎ = 16 are presented in Figure 4Under the limiting condition of 11986601 = 11986602 = infin (withoutporous walls) the numerical results produced by the study of[7] are also plotted for comparison Again good agreementsbetween the present analytical solutions and the publishednumerical results can be noticed The results in Figure 4(a)
Mathematical Problems in Engineering 11
YHZheng et al(2004)
963 100 87521 114kh
0
02
04
06
08
1
12
14m
2
as
G=G=infin
G=G=
G=G=
G=G=
(a) Added mass
YH Zhang et al(2004)
G=G=infin
G=G=
G=G=
G=G=
0010203040506070809
d
2
as
963 100 87521 114kh
(b) Radiation damping
Figure 5 Variation of the dimensionless pitch-pitch hydrodynamic coefficients ((a) for the added mass and (b) for the radiation damping)versus 1198960ℎ under the conditions of different porous-effect parameters for 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16 with the rotational center set at (00)
suggest that the variations of added mass for the shorterwavelength waves (large 1198960ℎ values) are less sensitive to thechange of porous parameters The presence of dual porouswalls increases the heave-heave added mass which tends toreach its maximum value at the long wave extreme Alsoa decrease in porous parameters results in an increase inheave induced added mass As shown in Figure 4(b) theheave-heave damping coefficients decrease rapidly when 1198960ℎincreases initially Then a gradual changing trend is followedby a further increase in 1198960ℎ Additionally the presence ofdual porous walls tends to reduce the values of heave-heavedamping coefficients
For the pitching induced added mass and dampingcoefficients the results are plotted against 1198960ℎ in Figure 5for various porous-effect parameters 11986601 and 11986602 and underthe settings of 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16 Whenthe rotational center is set at (00) as shown in Figure 5 thepresent analytical solutions match well with the publishednumerical results [7] for the limiting case of no porouswall (ie 11986601 = 11986602 = infin) It can be observed thatthe pitching oscillation induced added mass and dampingcoefficients for the floating breakwater system with dualporous side walls are greater than those without porous wallsWith decrease of the porous effect parameters the addedmass and damping coefficients increase For the added masscoefficients it is found that the variations trends are morecomplex for waves with relatively smaller 1198960ℎ under theconditions of different porous-effect parameters Generallyin cases of weaker porous-effect (for example 11986601 = 11986602 le 2as shown in Figure 5(a)) the value of added mass coefficienttends to approach 14 under the case considered For thevariations of damping coefficient as shown in Figure 5(b) thevarying trends are similar to those of damping coefficient insurge motion
The results presented in Figure 6 are the variations ofthe dimensionless surge-pitch added mass and damping
coefficients versus 1198960ℎ for varying 11986601 and 11986602 The case isfor 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16 with referenceto the rotational center of (00) Clearly the added massand damping coefficients are shown to increase with anincrease in the porous-effect parameters of both porouswalls A and B while the damping coefficients reach theirabsolute maximum values when 1198960ℎ is within the rangeof 1 and 2 The magnitude of surge-pitch hydrodynamiccoefficients especially the damping is greatly influenced bythe permeability of the dual porous walls compared to thesituation with surge or pitching motion only
The analytical solutions for the Response AmplitudeOperators (RAOs) of the floating breakwater system pro-posed in this study are illustrated in Figure 7 as a functionof dimensionless wavelength 1198960ℎ for various porous-effectparameters with the conditions of 119904ℎ = 03 119889ℎ = 05119886ℎ = 025 119903119866119886 = 09 and 119866119872119886 = 01 where therotational center is located at (00) It is found that the varyingtrends of surge RAOs are similar to those of pitching RAOsalthough the peak values of surge RAOs are smaller thanthose of pitching RAOs When the porous-effect parametersdecrease the main peak values of either surge or pitchingRAOs that occurred are shifted to the wave conditions withshorter wavelength Furthermore for the heave RAO itexperiences a typical damped response for the case of singledegree of freedom system and its maximum values of RAOwould move to the lower frequency wave region when theporous-effect parameters decrease
5 Summary and Conclusions
The hydrodynamic performance investigations includingthe hydrodynamic loads radiation induced added massand damping coefficients and dynamic responses relatedRAOs are critical to the study that is associated with theproblems of wave and structure interactions In this study the
12 Mathematical Problems in Engineering
G=G=infin
G=G=
G=G=
G=G=
963 100 87521 114kh
minus4
minus3
minus2
minus1
0
1
m
2
as
(a) Added mass
G=G=infin
G=G=
G=G=
G=G=
963 100 87521 114kh
minus3
minus25
minus2
minus15
minus1
minus05
0
d
2
as
(b) Radiation damping
Figure 6 Variation of the dimensionless surge-pitch hydrodynamic coefficients ((a) for the added mass and (b) for the radiation damping)versus 1198960ℎ under the conditions of different porous-effect parameters for 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16 with the rotational center set at(0 0)
G=G=
G=G=
G=G=
G=G=infin
4515 3525 3 605 2 5 551 4kh
0123456789
1011
|X|A
(a) Surge RAO
G=G=infin
G=G=
G=G=
G=G=
3 60 4 51 2kh
0
05
1
15
2
25
3
35|X
|A
(b) Heave RAO
G=G=
G=G=
G=G=
G=G=infin
05 3515 325 45 65 5541 20kh
0
5
10
15
|X|A
(c) Pitch RAO
Figure 7 Variation of the dimensionless RAO ((a) for surge RAO (b) for heave RAO and (c) for pitch RAO) versus 1198960ℎ under the conditionsof different porous-effect parameters for 119904ℎ = 03 119889ℎ = 05 119886ℎ = 025 119903119866119886 = 09 119866119872119886 = 01 with rotation center at (00)
Mathematical Problems in Engineering 13
hydrodynamic problems considering a floating breakwatersystem that consists of a rectangular body and two verticalattached porous walls are investigated analytically based onthe two-dimensional potential flow theory The expressionsof the diffraction and radiation based velocity potentials andthe associated unknown coefficients for each subdomain arederived by the MEEM approach Subsequently the excitinghydrodynamic forces added mass and radiation dampingcoefficients in terms of surge heave and pitching motionsand the equation ofmotion for the floating breakwater systemare formulated and calculated for various case scenariosby varying the wave conditions and porous effect param-eters Considering the limiting case without porous wallsgood agreements between the present analytical solutionsand published numerical results are achieved This suggeststhat the present analytical model is demonstrated to be areliable engineering tool to provide reasonable estimationsfor the hydrodynamic performances of a floating breakwatersystem with dual side porous walls It is worthy to noticethat the wave loads on the floating structures especiallythe moments when referenced to the rotational center ofbottom center would be reduced with an increase in theporous-effect parameters of the porous walls Additionallythe existence of the dual porous walls could move the peaksof surge and pitching RAOs to the waves of higher frequencywhen the porous effect parameters decrease Overall it canbe concluded that a properly adopted porous-wall systemwhen attached to a floating body can effectively improveits hydrodynamic performance as a breakwater The analyt-ical approach developed in this study can also be furtherextended to assist the theoretical analysis for other similar butimproved floating breakwater systems
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was supported by ldquoFundamental Research Fundsfor the Central Universitiesrdquo (Grand No 017180224)
References
[1] I-H Cho ldquoTransmission coefficients of a floating rectangularbreakwater with porous side platesrdquo International Journal ofNaval Architecture and Ocean Engineering vol 8 no 1 pp 53ndash65 2016
[2] H Hu K-H Wang and A N Williams ldquoWave motion over abreakwater system of a horizontal plate and a vertical porouswallrdquo Ocean Engineering vol 29 no 4 pp 373ndash386 2001
[3] C C Mei and J L Black ldquoScattering of surface waves byrectangular obstacles in water of finite depthrdquo Journal of FluidMechanics vol 38 no 3 pp 499ndash511 1969
[4] J L Black C C Mei and M C G Bray ldquoRadiation andscattering of water waves by rigid bodiesrdquo Journal of FluidMechanics vol 46 no 1 pp 151ndash164 1971
[5] N Drimer Y Agnon and M Stiassnie ldquoA simplified analyticalmodel for a floating breakwater in water of finite depthrdquoAppliedOcean Research vol 14 no 1 pp 33ndash41 1992
[6] J-F Lee ldquoOn the heave radiation of a rectangular structurerdquoOcean Engineering vol 22 no 1 pp 19ndash34 1995
[7] Y H Zheng Y G You and Y M Shen ldquoOn the radiationand diffraction of water waves by a rectangular buoyrdquo OceanEngineering vol 31 no 8-9 pp 1063ndash1082 2004
[8] E D Christensen H B Bingham A P Skou Friis A K Larsenand K L Jensen ldquoAn experimental and numerical study offloating breakwatersrdquo Coastal Engineering Journal vol 137 pp43ndash58 2018
[9] E Koutandos P Prinos andXGironella ldquoFloating breakwatersunder regular and irregular wave forcing Reflection and trans-mission characteristicsrdquo Journal of Hydraulic Research vol 43no 2 pp 174ndash188 2005
[10] G H Dong Y N Zheng Y C Li B Teng C T Guan and D FLin ldquoExperiments on wave transmission coefficients of floatingbreakwatersrdquo Ocean Engineering vol 35 no 8-9 pp 931ndash9382008
[11] A T Chwang ldquoA porous-wavemaker theoryrdquo Journal of FluidMechanics vol 132 pp 395ndash406 1983
[12] Y C Li Y Liu and B Teng ldquoPorous effect parameter of thinpermeable platesrdquo Coastal Engineering Journal vol 48 no 4pp 309ndash336 2006
[13] S R Manam and M Sivanesan ldquoScattering of water wavesby vertical porous barriers an analytical approachrdquo WaveMotion An International Journal Reporting Research on WavePhenomena vol 67 pp 89ndash101 2016
[14] A T Chwang and Z Dong ldquoWave-trapping due to a porousplaterdquo in Proceedings of 15 ONR symposium on Naval Hydrody-namics pp 407ndash414 1985
[15] S Zhu and A T Chwang ldquoAnalytical study of porous waveabsorberrdquo Journal of Engineering Mechanics vol 127 no 4 pp326ndash332 2001
[16] H Behera and T Sahoo ldquoHydroelastic analysis of gravity waveinteraction with submerged horizontal flexible porous platerdquoJournal of Fluids and Structures vol 54 pp 643ndash660 2015
[17] T L Yip and A T Chwang ldquoPerforated wall breakwater withinternal horizontal platerdquo Journal of Engineering Mechanics vol126 no 5 pp 533ndash538 2000
[18] Y Liu and H-J Li ldquoWave scattering by dual submergedhorizontal porous plates Further resultsrdquo Ocean Engineeringvol 81 no 5 pp 158ndash163 2014
[19] I H Cho and M H Kim ldquoWave absorbing system usinginclined perforated platesrdquo Journal of Fluid Mechanics vol 608pp 1ndash20 2008
[20] K-H Wang and X Ren ldquoWater waves on flexible and porousbreakwatersrdquo Journal of Engineering Mechanics vol 119 no 5pp 1025ndash1047 1993
[21] K H Wang and X Ren ldquoWave interaction with a concentricporous cylinder systemrdquo Ocean Engineering vol 21 no 4 pp343ndash360 1994
[22] F Zhao T Zhang R Wan L Huang X Wang and WBao ldquoHydrodynamic loads acting on a circular porous platehorizontally submerged in wavesrdquo Ocean Engineering vol 136pp 168ndash177 2017
14 Mathematical Problems in Engineering
[23] A Mondal and R Gayen ldquoWave interaction with dual circularporous platesrdquo Journal of Marine Science and Application vol14 no 4 pp 366ndash375 2015
[24] W L Qiao K-H Wang and Y Q Sun ldquoScattering of waterwaves by a floating body with two vertically attached porouswallsrdquo Journal of Engineering Mechanics vol 144 no 2 ArticleID 04017162 2018
[25] S W Twu and D T Lin ldquoOn a highly effective wave absorberrdquoCoastal Engineering Journal vol 15 no 4 pp 389ndash405 1991
[26] X P Yu ldquoDiffraction of water waves by porous breakwatersrdquoJournal of Waterway Port Coastal and Ocean Engineering vol121 no 6 pp 275ndash282 1995
[27] I J Losada M A Losada and A Baquerizo ldquoAn analyticalmethod to evaluate the efficiency of porous screens as wavedampersrdquo Applied Ocean Research vol 15 no 4 pp 207ndash2151993
[28] G Besio and M A Losada ldquoSediment transport patterns atTrafalgar offshore windfarmrdquo Ocean Engineering vol 35 no 7pp 653ndash665 2008
[29] H Song and L Tao ldquoShort-crestedwave interactionwith a con-centric porous cylindrical structurerdquo Applied Ocean Researchvol 29 no 4 pp 199ndash209 2007
[30] L Xiao Y Kou L Tao and L Yang ldquoComparative studyof hydrodynamic performances of breakwaters with double-layered perforated walls attached to ring-shaped very largefloating structuresrdquo Ocean Engineering vol 111 pp 279ndash2912016
[31] T L Yip and A T Chwang ldquoWater wave control by submergedpitching porous platerdquo Journal of Engineering Mechanics vol124 no 4 pp 428ndash434 1998
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4 Mathematical Problems in Engineering
Here 119896119898 can be calculated from
1205962 = minus119892119896119898 tanh (119896119898ℎ) (13)
Substituting (7) (8) and (9) into the following derivedmatching equations
int0minusℎ
1205971206011198631120597119909 cos 119899 (119911 + ℎ) 119889119911= intminus119904minusℎ
1205971206011198632120597119909 cos 119899 (119911 + ℎ) 119889119911119909 = minus119886 (119899 = 0 1 2 3 )
(14)
int0minusℎ
1205971206011198633120597119909 cos 119899 (119911 + ℎ) 119889119911= intminus119904minusℎ
1205971206011198632120597119909 cos 119899 (119911 + ℎ) 119889119911119909 = minus119886 (119899 = 0 1 2 3 )
(15)
intminus119889minusℎ
1206011198631 cos 119903119899 (119911 + ℎ) 119889119911 + intminus119904minus119889
1198631 cos 119903119899 (119911 + ℎ) 119889119911= intminus119904minusℎ
1206011198632 cos 119903119899 (119911 + ℎ) 119889119911119909 = minus119886 (119899 = 0 1 2 )
(16)
intminus119889minusℎ
1206011198633 cos 119903119899 (119911 + ℎ) 119889119911 + intminus119904minus119889
1198633 cos 119903119899 (119911 + ℎ) 119889119911= intminus119904minusℎ
1206011198632 cos 119903119899 (119911 + ℎ) 119889119911119909 = minus119886 (119899 = 0 1 2 )
(17)
and utilizing the orthogonality properties of solutioneigenfunctions the unknown coefficients 119862119899 119863119899 119877119899 119879119899 (119899 =0 1 2 3 119872) for the diffraction potentials can be deter-mined by solving the 4(M+1) system of algebraic equationsHere the terms with summation of infinite series are approx-imated by a large but finite number of M terms In (16) and(17) 1198631 and 1198633 are given as
1198631 = infinsum119898=0
[(119868119898 + 119865119898) minus 119898i119866011198960 (119868119898 minus 119865119898)]
sdot cos 119898 (119911 + ℎ)cos 119898ℎ
(18)
1198633 = infinsum119898=0
(1 + 119898i119866011198960) 119879119898 cos 119898 (119911 + ℎ)
cos 119898ℎ (19)
where 11986601 = 1205881205961198871(1205831198960) and 11986602 = 1205881205961198872(1205831198960) aredimensionless parameters defined according to Chwang [11]to represent the porous effect parameters of porous walls Aand B respectively
22 eoretical Formulations of the Radiated Velocity Poten-tials For the radiation problem the displacements of thefloating breakwater system in motion are given as
119883119895 = 1198830119895119890minusi120596119905 119895 = 1 2 3 (20)
where 119883119900119895 (119895 = 1 2 3) are the amplitudes of surge heaveand pitching motions respectively Let 120601(119871)119877119894 (119871 = 1 2 3 119894 =1 2 3) be the spatial radiated potentials describing the small-amplitude oscillations where 119871119905ℎ indices denote respectivelythe induced surge heave and pitching motions while the 119894119905ℎindices represent the subdomains of I II and III respectivelyThen the total radiated potentials in any of the flow domainscan be expressed as
Φ(119871)119877 = 119883119895120601(119871)119877 = Re [minusi1205961198830119895120601(119871)119877 (119909 119911) 119890minusi120596119905] (21)
As the velocity potentials120601(119871)119877 satisfy the Laplace equationwe have
nabla2120601(119871)119877119894 = 0 (119871 = 1 2 3 119894 = 1 2 3) (22)
The boundary conditions including the free surfaceboundary conditions bottom boundary conditions and thebody surface conditions for the surge heave and pitchingoscillations are listed as follows respectively
120597120601(1)119877119894120597119911 = 0 (119894 = 1 2 3) 119911 = ℎ1205972120601(1)1198771198941205971199052 + 120597120601(1)119877119894120597119911 = 0 (119894 = 1 3) 119911 = 0 |119909| ge 119886
120597120601(1)1198772120597119911 = 0 119911 = minus119904 |119909| le 119886(23)
120597120601(2)119877119894120597119911 = 0 (119894 = 1 2 3) 119911 = ℎ1205972120601(2)1198771198941205971199052 + 119892 120597120601(2)119877119894120597119911 = 0 (119894 = 1 3) 119911 = 0 |119909| ge 119886
120597120601(2)1198772120597119911 = 1 119911 = minus119904 |119909| le 119886(24)
120597120601(3)119877119894120597119911 = 0 (119894 = 1 2 3) 119911 = ℎ1205972120601(3)1198771198941205971199052 + 119892 120597120601(1)119877119894120597119911 = 0 (119894 = 1 3) 119911 = 0 |119909| ge 119886
120597120601(3)119877119894120597119899 = i120596 [(119911 minus 120585119911) 119899119909 minus (119909 minus 120585119909) 119899119911]119900119899 119887119900119889119910 119904119906119903119891119886119888119890
(25)
where 119899 = (119899119909 119899119911) is the unit normal vector and (120585119909 120585119911)is the rotational center of the pitching oscillation With theassumption that the rotational center is along the z axis wecan get 120585119909 = 0
Mathematical Problems in Engineering 5
23 Analytical Solutions of the Radiated Velocity PotentialsSimilar to the derivation of the solutions of the diffractedvelocity potentials the application of the method of sepa-ration variables to the governing equation (22) under theboundary conditions (23) (24) and (25) yields the radiatedvelocity potentials as follows
120601(119871)1198771 = infinsum119898=0
119860(119871)119898 119890minus119896119898(119909+119886) cos 119898 (119911 + ℎ)cos 119898ℎ 119890minusi120596119905 (26)
120601(1)1198772 = (119862(1)0 119909 + 119864(1)0 ) 119890minusi120596119905+ infinsum119898=1
[119862(1)119898 cosh (119903119898119909)cosh (119903119886119909) + 119864(1)119898 sinh (119903119898119909)
sinh (119903119886119909) ]sdot cos 119903119898 (119911 + ℎ) 119890minusi120596119905
(27)
120601(2)1198772 = (119911 + ℎ)2 minus 11990922 (ℎ minus 119904) + (119862(2)0 119909 + 119864(2)0 ) 119890minusi120596119905
+ infinsum119898=1
[119862(2)119898 cosh (119903119898119909)cosh (119903119886119909) + 119864(2)119898 sinh (119903119898119909)
sinh (119903119886119909) ]sdot cos 119903119898 (119911 + ℎ) 119890minusi120596119905
(28)
120601(3)1198772 = infinsum119898=1
120576 cos [120574119898 (119909 minus 119886)] cosh [120574119898 (119911 + ℎ)]+ (119862(3)0 119909 + 119864(3)0 ) 119890minusi120596119905+ infinsum119898=1
[119862(3)119898 cosh (119903119898119909)cosh (119903119886119909) + 119864(3)119898 sinh (119903119898119909)
sinh (119903119886119909) ]sdot cos 119903119898 (119911 + ℎ) 119890minusi120596119905
(29)
120601(119871)1198773 = infinsum119898=0
119861(119871)119898 119890119896119898(119909minus119886) cos 119898 (119911 + ℎ)cos 119898ℎ 119890minusi120596119905 (30)
where
120574119898 = 1198981205872119886 (119898 = 1 2 3 ) (31)
Combining the floating body surface condition at 119911 = minus119904minus119886 le 119909 le 119886 and the orthogonality of cos[120574119898(119909 minus 119886)] over theintegral interval of minus119886 le 119909 le 119886 the expression of 120576 can bederived as
120576 = cos 2119886120574119898sinh [120574119898 (ℎ minus 119904)] 1198861205741198983 (32)
24 Determination of theUnknownCoefficientsatAppearedin the Solutions of Radiated Velocity Potentials The solutionsof the unknown coefficients that appeared in the radiationpotentials can be again obtained by utilizing the matchingconditions at 119909 = plusmn119886 under the principle of continuouspressure and normal velocity We have the following
(a) Matching Conditions for the Surge-Induced Potential
120601(1)1198772 = 120601(1)1198771 minusℎ le 119911 le minus119889 119909 = minus119886120601(1)1198773 minusℎ le 119911 le minus119889 119909 = 119886 (33a)
120597120601(1)1198771120597119909 = 120597120601(1)1198772120597119909 minusℎ le 119911 le minus119889 119909 = minus1198861 minus119904 le 119911 le 0 119909 = 119886 (33b)
120597120601(1)1198773120597119909 = 120597120601(1)1198772120597119909 minusℎ le 119911 le minus119889 119909 = minus1198861 minus119904 le 119911 le 0 119909 = 119886 (33c)
(b) Matching Conditions for the Heave-Induced Potential
120601(2)1198772 = 120601(2)1198771 minusℎ le 119911 le minus119889 119909 = minus119886120601(2)1198773 minusℎ le 119911 le minus119889 119909 = 119886 (34a)
120597120601(2)1198771120597119909 = 120597120601(2)1198772120597119909 minusℎ le 119911 le minus119889 119909 = minus1198861 minus119904 le 119911 le 0 119909 = 119886 (34b)
120597120601(2)1198773120597119909 = 120597120601(2)1198772120597119909 minusℎ le 119911 le minus119889 119909 = minus1198861 minus119904 le 119911 le 0 119909 = 119886 (34c)
(c) Matching Conditions for the Pitching-Induced Potential
120601(3)1198772 = 120601(3)1198771 minusℎ le 119911 le minus119889 119909 = minus119886120601(3)1198773 minusℎ le 119911 le minus119889 119909 = 119886 (35a)
120597120601(3)1198771120597119909 = 120597120601(3)1198772120597119909 minusℎ le 119911 le minus119889 119909 = minus119886119911 minus 120585119911 minus119904 le 119911 le 0 119909 = 119886 (35b)
120597120601(3)1198773120597119909 = 120597120601(3)1198772120597119909 minusℎ le 119911 le minus119889 119909 = minus119886119911 minus 120585119911 minus119904 le 119911 le 0 119909 = 119886 (35c)
For the porous walls A and B at the boundaries 119909 =plusmn119886 the condition of continuity of normal velocity for flowspassing through the porous walls A and B leads to
120597120601(119871)1198771120597119909 = 120597120601(119871)1198772120597119909 = 119882119860 (119911 119905)119871 = (1 2 3) minus ℎ le 119911 le minus119889 119909 = minus119886
(36a)
120597120601(119871)1198773120597119909 = 120597120601(119871)1198772120597119909 = 119882119861 (119911 119905)119871 = (1 2 3) minus ℎ le 119911 le minus119889 119909 = minus119886
(36b)
where the porous flow velocities 119882119860(119911 119905) and 119882119861(119911 119905) areassumed to obey Darcys law that is they are linearly pro-portional to the pressure difference induced by the radiated
6 Mathematical Problems in Engineering
motions across their corresponding porous wall (Chwang[11]) With the application of the linearized Bernoulli equa-tion we have
119882119860 (119911 119905) = minus11986601 [ 1198960120596 (120597120601(119871)1198771120597119905 minus 120597120601(119871)1198772120597119905 )](119871 = 1 2 3) 119909 = minus119886 minus119889 le 119911 le minus119904
(37a)
119882119861 (119911 119905) = minus11986602 [ 1198960120596 (120597120601(119871)1198772120597119905 minus 120597120601(119871)1198773120597119905 )](119871 = 1 2 3) 119909 = 119886 minus119889 le 119911 le minus119904
(37b)
Utilizing the orthogonality property of eigenfunctionscos 119899(119911 + ℎ) for 120601(119871)1198771 and 120601(119871)1198773 (119871 = 1 2 3) over the integralinterval of minusℎ le 119911 le 0 the following equations can beformulated by the matching conditions of (33b) (33c) (34b)(34c) (35b) and (35c) as
int0minusℎ
120597120601(1)1198771120597119909 cos 119899 (119911 + ℎ) 119889119911= intminus119904minusℎ
120597120601(1)1198772120597119909 cos 119899 (119911 + ℎ) 119889119911+ int0minus119904
1 cos 119899 (119911 + ℎ) 119889119911 (119909 = minus119886)(38)
int0minusℎ
120597120601(1)1198773120597119909 cos 119899 (119911 + ℎ) 119889119911= intminus119904minusℎ
120597120601(1)1198772120597119909 cos 119899 (119911 + ℎ) 119889119911+ int0minus119904
1 cos 119899 (119911 + ℎ) 119889119911 (119909 = 119886)(39)
int0minusℎ
120597120601(2)1198771120597119909 cos 119899 (119911 + ℎ) 119889119911= intminus119904minusℎ
120597120601(2)1198772120597119909 cos 119899 (119911 + ℎ) 119889119911 (119909 = minus119886)(40)
int0minusℎ
120597120601(2)1198773120597119909 cos 119899 (119911 + ℎ) 119889119911= intminus119904minusℎ
120597120601(2)1198772120597119909 cos 119899 (119911 + ℎ) 119889119911 (119909 = 119886)(41)
int0minusℎ
120597120601(3)1198771120597119909 cos 119899 (119911 + ℎ) 119889119911= intminus119904minusℎ
120597120601(3)1198772120597119909 cos 119899 (119911 + ℎ) 119889119911+ int0minus119904
(119911 minus 120585119911) cos 119899 (119911 + ℎ) 119889119911 (119909 = minus119886)(42)
int0minusℎ
120597120601(3)1198773120597119909 cos 119899 (119911 + ℎ) 119889119911
= intminus119904minusℎ
120597120601(3)1198772120597119909 cos 119899 (119911 + ℎ) 119889119911+ int0minus119904
(119911 minus 120585119911) cos 119899 (119911 + ℎ) 119889119911 (119909 = 119886)
(43)
Additionally by combining (36a) (36b) (37a) and (37b)with the matching conditions given in (33a) (34a) and(35a) respectively and applying the orthogonality propertyof eigenfunctions cos 119903119899(119911 + ℎ) for 120601(119871)1198772 (L = 1 2 3) over theintegral interval of minusℎ le 119911 le minus119904 we have
intminus119904minusℎ
120601(119871)1198772 cos 119903119899 (119911 + ℎ) 119889119911= intminus119889minusℎ
120601(119871)1198771 cos 119903119899 (119911 + ℎ) 119889119911+ intminus119904minus119889
(119871)1198771 cos 119903119899 (119911 + ℎ) 119889119911(119909 = minus119886 119871 = 1 2 3)
(44)
intminus119904minusℎ
120601(119871)1198772 cos 119903119899 (119911 + ℎ) 119889119911= intminus119889minusℎ
120601(119871)1198773 cos 119903119899 (119911 + ℎ) 119889119911+ intminus119904minus119889
(119871)1198773 cos 119903119899 (119911 + ℎ) 119889119911(119909 = 119886 119871 = 1 2 3)
(45)
where
(1)1198771 = infinsum119898=0
119860(1)119898 (1 + 119898i119866011198960)
cos 119898 (119911 + ℎ)cos 119898ℎ
+ 1i119866011198960
(46)
(2)1198771 = infinsum119898=0
119860(2)119898 (1 + 119898i119866011198960)
cos 119898 (119911 + ℎ)cos 119898ℎ (47)
(3)1198771 = infinsum119898=0
119860(3)119898 (1 + 119898i119866011198960)
cos 119898 (119911 + ℎ)cos 119898ℎ
+ 119911i119866011198960
(48)
Mathematical Problems in Engineering 7
(1)1198773 = infinsum119898=0
119861(1)119898 (1 + 119898i119866021198960)
cos 119898 (119911 + ℎ)cos 119898ℎ
minus 1i119866021198960
(49)
(2)1198773 = infinsum119898=0
119861(2)119898 (1 minus 119898i119866021198960)
cos 119898 (119911 + ℎ)cos 119898ℎ (50)
(3)1198773 = infinsum119898=0
119861(3)119898 (1 + 119898i119866021198960)
cos 119898 (119911 + ℎ)cos 119898ℎ
minus 119911i119866021198960
(51)
Again the summation of terms with the infinite seriesis approximated by a large but finite numbers of termseg a total of M terms Then the unknown coefficients119860(1)119898 119862(1)119898 119864(1)119898 119861(1)119898 (119898 = 0 1 2 119872) [4(M+1) unknownstotally] can be determined by solving the 4(M+1) sys-tem of algebraic equations formulated in (38) (39) (44)and (45) (for L=1) Similarly the unknown coefficients119860(2)119898 119862(2)119898 119864(2)119898 119861(2)119898 (119898 = 0 1 2 119872) can be calculatedfrom the 4(M+1) system of algebraic equations expressedin (40) (41) (44) and (45) (for L=2) while the coefficients119860(3)119898 119862(3)119898 119864(3)119898 119861(3)119898 (119898 = 0 1 2 119872) can be evaluated bysolving the 4(M+1) system of algebraic equations formulatedin (42) (43) (44) and (45) (for L=3) After the determinationof the above unknown coefficients a series of engineeringproperties important to practical applications could be cal-culated for example the hydrodynamic coefficients and themotion induced dynamic responses of the breakwater system
3 Exciting Forces andHydrodynamic Coefficients
31 Exciting Forces To determine the hydrodynamic coeffi-cients related to the inducedmotions of the floating breakwa-ter system the exciting forces and moment of the combinedincident and diffracted waves on the breakwater need tobe computed as inputs for solving the equation of motionBy integrating the dynamic pressures with the use of thevelocity potentials of incident and diffracted waves (see (7)-(9)) over the wet surface the hydrodynamic forces alongthe x direction (1198651198631) and along the z direction (1198651198632) can beformulated respectively as (Qiao et al [24])
1198651198631= i120588120596 infinsum
119898=0
(119868119898 + 119865119898 minus 119879119898) [sin 119898ℎ minus sin 119898 (ℎ minus 119889)]119898 cos 119898ℎ
sdot 119890minusi120596119905 + i120588120596 lowast 2119886 (119889 minus 119904) 1198620119890minusi120596119905minus i120588120596 infinsum
119898=0
2119864119898119903119898 sin 119903119898 (ℎ minus 119889) 119890minusi120596119905(52)
1198651198632 = i120596120588 [21198861198640 + 2 infinsum119898=1
119862119898119903119898 tanh (119903119898119886) cos 119903119898 (ℎ minus 119904)]sdot 119890minusi120596119905
(53)
Also the wave exciting moment (1198651198633) that is referencedto a rotational center (0120585119911) can be derived as
1198651198633 = i120588120596
(119868119898 + 119865119898 minus 119879119898)
sdot [[[[[[
119889 sin 119898 (ℎ minus 119889) minus 120585119911 sin 119898ℎ + 120585119911 sin 119898 (ℎ minus 119889)119898 cos 119898ℎ
+cos 119898ℎ minus cos 119898 (ℎ minus 119889)1198982 cos 119898ℎ
]]]]]]
sdot 119890minusi120596119905 + i120588120596 1198861198620 (1198892 minus 1199042)
minus infinsum119898=1
2119864119898119903119898 [119889 sin 119903119898 (ℎ minus 119889)
+ cos 119903119898 (ℎ minus 119904) minus cos 119903119898 (ℎ minus 119889)119903119898 ] 119890minusi120596119905
+ i120588120596 [21198861198620120585119911 (119889 minus 119904) minus infinsum119898=1
2119864119898120585119911119903119898 sin 119903119898 (ℎ minus 119889)]
sdot 119890minusi120596119905 minus i120588120596 2119862011988633 + infinsum119898=1
[ 2119886119864119898 cos 119903119898 (ℎ minus 119904)119903119898 tanh (119903119898119886)minus 2119864119898 cos 119903119898 (ℎ minus 119904)1199031198982 ] 119890minusi120596119905
(54)
32 Hydrodynamic Coefficients The radiation forces inducedby the translational and rotational motions of the floatingbreakwater system can be calculated from the derived radi-ation potentials utilizing the linearized Bernoulli equation as
119865119871119896 = i120588120596 intΓ
Φ(119871)119877119896 119890minusi120596119905119899119895119889Γ= 12058812059621198830119895 int
Γ120601(119871)119877119896 119890minusi120596119905119899119895119889Γ
= 12059621198830119895119898119871119896119890minusi120596119905 + i1205961198830119895119889119871119896119890minusi120596119905(55)
where 119898119871119896 is the radiation-induced added mass and the119889119871119896 denotes the radiation damping The expressions arerespectively
119898119871119896 = Re(120588 intΓ
120601(119871)119877119896119899119895119889Γ) = Re (120588119871119896) (56)
119889119871119896 = Im (120588120596 intΓ
120601(119871)119877119896119899119895119889Γ) = Im (120588120596119871119896) (57)
8 Mathematical Problems in Engineering
119871119896 can be computed with formulations given below
11 = (int0minus119889
120601(1)1198771 10038161003816100381610038161003816119909=minus119886 119889119911 minus int0minus119889
120601(1)1198773 10038161003816100381610038161003816119909=119886 119889119911)+ (intminus119904minus119889
120601(1)1198772 10038161003816100381610038161003816119909=minus119886 119889119911 minus intminus119904minus119889
120601(1)1198772 10038161003816100381610038161003816119909=119886 119889119911)= 119872sum119898=1
119860(1)119898 minus 119861(1)119898119898 cos 119898ℎ [sin 119898ℎ minus sin 119898 (ℎ minus 119889)]+ 2119886119862(1)0 (119889 minus 119904) + 119872sum
119898=1
2119864(1)119898119903119898 [sin119898120587minus sin 119903119898 (ℎ minus 119889)]
(58)
22 = int119886minus119886
120601(2)1198772 10038161003816100381610038161003816119911=minus119909 119889119909 = 2119886119864(2)0+ 119872sum119898=1
2119862(2)119898119903119898 tanh (119903119898119886) cos 119903119898 (ℎ minus 119904) + 119886 (ℎ minus 119904)minus 11988633 (ℎ minus 119904)
(59)
33 = int0minus119889
( 120601(3)1198771 10038161003816100381610038161003816119909=minus119886 minus 120601(3)1198773 10038161003816100381610038161003816119909=119886) (119911 minus 120585119911) 119889119911+ intminus119904minus119889
(120601(3)1198772 10038161003816100381610038161003816119909=minus119886 minus 120601(3)1198772 10038161003816100381610038161003816119909=119886) (119911 minus 120585119911) 119889119911minus int119886minus119886
120601(3)1198772 10038161003816100381610038161003816119909=minus119904 119909 119889119909 = 119872sum119898=0
(119860(3)119898 minus 119861(3)119898 )
sdot [119889 sin 119898 (ℎ minus 119889) minus 120585119911 sin 119898 (ℎ minus 119889)119898 cos 119898ℎ
+ cos 119898ℎ minus cos 119898 (ℎ minus 119889)1198982 cos 119898ℎ ] + 119886119862(3)0 (1198892 minus 1199042)
minus 119872sum119898=1
2119864(3)119898119903119898 [119889 sin 119898 (ℎ minus 119889)+ cos 119903119898 (ℎ minus 119904) minus cos 119903119898 (ℎ minus 119889)119903119898 ] + 2119886119862(3)0 120585119911 (119889minus 119904) minus 119872sum
119898=1
2119864(3)119898 120585119911119903119898 sin 119903119898 (ℎ minus 119889) minus 23 119862(3)0 1198863 minus 119872sum119898=1
120576sdot cosh 120574119898 (ℎ minus 119904) 1 minus cos 21198861205741198981205741198982 + 119872sum
119898=1
(120576 cos119898120587minus 120576) [ 119889 sinh 120574119898 (ℎ minus 119889) minus 119904 sinh 120574119898 (ℎ minus 119904)120574119898minus cosh 120574119898 (ℎ minus 119904) minus cosh 120574119898 (ℎ minus 119889)1205741198982 ]minus 119872sum119898=1
119864(3)119898 cos 119903119898 (h minus s)119903119898 sinh 119903119898119886 (2119886 cosh 119903119898119886minus 2 sinh 119903119898119886119903119898 )
(60)
31 = 13 = int0minus119889
( 120601(3)1198771 10038161003816100381610038161003816119909=minus119886 minus 120601(3)1198773 10038161003816100381610038161003816119909=119886) 119889119911+ intminus119904minus119889
( 120601(3)1198772 10038161003816100381610038161003816119909=minus119886 minus 120601(3)1198772 10038161003816100381610038161003816119909=119886) 119889119911 = 119872sum119898=0
(119860(3)119898minus 119861(3)119898 ) sin 119898ℎ minus sin 119898 (ℎ minus 119889)
119898 cos 119898ℎ minus 2 (ℎ minus 119904)sdot [ 11988636 (ℎ minus 119904) + 119886119862(3)0 ] + 119886 (ℎ minus 119904)3 minus 119886 (ℎ minus 119889)33 (ℎ minus 119904)+ 119872sum119898=1
2119864(3)119898119903119898 sin 119903119898 (ℎ minus 119889)
(61)
33 Equation of Motion and RAOs The derived solutions forthe diffraction and radiation problems could now be inte-grated with the equation of motion of the floating breakwatersystem to analyze the dynamic responses of the structuralsystem We have
minus1205962 [119872 + 119898] minus i120596 [119889] + [119904] [1198830] = [1198650] (62)
where [1198830] = [11988301 11988302 11988303]119879 and are vectors indicatingthe motion amplitudes and the wave exciting forces respec-tively [119898] and [119889] are addedmass and dampingmatrices [119872]is the mass matrix and [119878] denotes the hydrostatic restoringforce matrix they can be expressed as
[119872] = [[[
1198980 0 minus11989801198851198660 1198980 0minus1198980119885119866 0 1]]]
(63)
[119878] = [[[
0 0 00 2120588119892119886 00 0 1198980119892119866119872
]]]
(64)
Here 1198980 is the mass per unit length of the floatingbreakwater and 119911119866 is the z-coordinate of the center of gravityThe mass moment of inertia about y-axis I can be computedby 119868 = 11989801199031198662 in which 119903119866 is the radius of gyration in pitchingoscillation and 119866119872 is the metacentric height of the floatingbreakwater system Once 1198830119895 (j=1 2 3) are determined bysolving (62) the RAOs (Response Amplitude Operators) ofthe breakwater system can be calculated as
119877119860119874119895 = 1198830119895119867119895 (119895 = 1 2 3) (65)
4 Results and Discussion
The validation of the wave diffraction solutions in terms ofreflectiontransmission coefficients and hydrodynamic loadsfor a train of monochromatic wave propagating throughthe proposed floating breakwater system that consists of apartially submerged rectangular body and with or withoutthe attached two porous side walls has been carried outin a previous study (Qiao et al [24]) Here additional
Mathematical Problems in Engineering 9
Black et al(1971)Present approach
G=G=infinhs=as=
0010203040506070809
F D
gH
a
3 60 5 71 42ks
(a) Horizontal force
Black et al(1971)Present approach
G=G=infinhs= as=
0010203040506070809
1
F D
gHa
3 60 5 71 42ks
(b) Vertical force
Black et al(1971)Present approach
G=G=infinhs= as=
3 60 4 51 72ks
0
005
01
015
02
025
F D
gH
a2
(c) Moment
Figure 2 Comparison of the dimensionless wave exciting forces and moment versus 1198960119904 for ℎ119904 = 20 119886119904 = 30 11986601 = 11986602 997888rarr infin (a)dimensionless horizontal exciting force (b) dimensionless vertical exciting force (c) dimensionless exciting moment about the rotationalcenter of (0 0)
comparisons of wave diffraction results with other publishednumerical solutions for the limiting case of neglecting theeffect of two porous walls are again presented For thewave radiation problem the motion induced hydrodynamiccoefficients such as added mass and damping coefficient arealso investigated for their variations with changes of waveparameters As a validation those results for the caseswithoutporous walls are compared with limited numerical solutionspublished in literatures Subsequently the effects of variousdimensionless parameters for example porous propertiesincident wave conditions and the geometry of floating bodyto wave-induced hydrodynamic forces hydrodynamic coeffi-cients and RAOs were also examined In our computationsthe first 40 terms in the infinite series of the diffractionpotentials and radiation potentials are taken the same as [7]namely M=40
As an example showing the results comparisons for thediffraction problem Figure 2 illustrates the variations ofthe dimensionless exciting forces in horizontal and verticaldirections versus dimensionless wavelength 1198960119904 for the caseof ℎ119904 = 20 119886119904 = 10 11986601 = 11986602 997888rarr infin The excitingmoments with reference to the rotational center of (00) are
also presented in Figure 2 Here 11986601 = 11986602 997888rarr infin representsthe case of a floating breakwater system without dual porousside walls The numerical results computed by Black et al [4]are also plotted in Figure 2 The comparisons indicate thatthe exciting forces and moments are well predicted by thepresent analytical model Both the analytical solutions andthe numerical results reveal that the horizontal exciting forcesand the exciting moments show rapidly increasing trendinitially until a peak value is reached and then are followedby a gradual decrease of the values with the further increaseof 1198960119904 However the vertical forces as shown in Figure 2(b)decrease continuously with an increase in 1198960119904 According tothe results given in the previous study by Qiao et al [24] itindicated that the effectiveness of the porous walls is reflectedwith the reduction of the vertical hydrodynamic forces andwhen the breakwater system is rotated with reference to thebottom center of the body the induced moment can also bereduced However due to the added additional length of theporous walls the horizontal forces are slightly greater thanthose without porous walls
Figure 3 presents the variations of the dimensionlessadded mass and damping coefficients in surge only motion
10 Mathematical Problems in Engineering
YHZheng et al(2004)
G=G=
G=G=
G=G=
G=G=infin
0
1
2
3
4
5
6m
2
as
963 100 87521 114kh
(a) Added mass
YH Zheng et al(2004)
G=G=infin
G=G=
G=G=
G=G=
963 100 87521 114kh
005
115
225
335
4
d
2
as
(b) Radiation coefficient
Figure 3 Variation of the dimensionless surge-surge hydrodynamic coefficients ((a) for the added mass and (b) for the radiation damping)versus 1198960ℎ under the conditions of different porous-effect parameters for 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16
YHZheng et al(2004)
G=G=
G=G=
G=G=
G=G=infin
963 100 87521 114kh
0
02
04
06
08
1
12
14
m
2
as
(a) Added mass
YHZheng et al(2004)
G=G=
G=G=
G=G=
G=G=infin
minus04minus02
002040608
11214
d
2
as
963 100 87521 114kh
(b) Radiation damping
Figure 4 Variation of the dimensionless heave-heave hydrodynamic coefficients ((a) for the added mass and (b) for the radiation damping)versus 1198960ℎ under the conditions of different porous-effect parameters for 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16
versus 1198960ℎ for the setting of 119904ℎ = 13 119889ℎ = 12 and 119886ℎ =16under the conditions of various porous-effect parametersWhen 11986601 = 11986602 = infin it represents the case withoutconsidering the effect of dual porous walls The presentanalytical solutions for this limiting case can be adopted tocompare to the limited numerical results available in thestudy of [7] The comparison plot is shown in Figure 4 wherethe nearly identical matches between the present analyticalsolutions and the numerical results of [7] can be observedThe results shown in Figure 4 suggest that the addition ofporous walls tends to enhance the added mass and dampingcoefficients In general with a decrease of porous effectparameter the addedmass and damping coefficients increaseFor the variation of addedmass coefficient with porouswalls
it can be seen that a peak value occurred at a long wave (small1198960ℎ value) condition However for the radiation dampingcoefficient it is shown to increase rapidly to its maximumvalue with an increase of 1198960ℎ then a gradual decrease trendis followed by a further increase of 1198960ℎ
The dimensionless heave-heave added mass (119898222120588119886119904)and radiation damping coefficient (119889222120588120596119886119904) versus 1198960ℎunder various 11986601 and 11986602 values for the case of 119904ℎ =13 119889ℎ = 12 119886ℎ = 16 are presented in Figure 4Under the limiting condition of 11986601 = 11986602 = infin (withoutporous walls) the numerical results produced by the study of[7] are also plotted for comparison Again good agreementsbetween the present analytical solutions and the publishednumerical results can be noticed The results in Figure 4(a)
Mathematical Problems in Engineering 11
YHZheng et al(2004)
963 100 87521 114kh
0
02
04
06
08
1
12
14m
2
as
G=G=infin
G=G=
G=G=
G=G=
(a) Added mass
YH Zhang et al(2004)
G=G=infin
G=G=
G=G=
G=G=
0010203040506070809
d
2
as
963 100 87521 114kh
(b) Radiation damping
Figure 5 Variation of the dimensionless pitch-pitch hydrodynamic coefficients ((a) for the added mass and (b) for the radiation damping)versus 1198960ℎ under the conditions of different porous-effect parameters for 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16 with the rotational center set at (00)
suggest that the variations of added mass for the shorterwavelength waves (large 1198960ℎ values) are less sensitive to thechange of porous parameters The presence of dual porouswalls increases the heave-heave added mass which tends toreach its maximum value at the long wave extreme Alsoa decrease in porous parameters results in an increase inheave induced added mass As shown in Figure 4(b) theheave-heave damping coefficients decrease rapidly when 1198960ℎincreases initially Then a gradual changing trend is followedby a further increase in 1198960ℎ Additionally the presence ofdual porous walls tends to reduce the values of heave-heavedamping coefficients
For the pitching induced added mass and dampingcoefficients the results are plotted against 1198960ℎ in Figure 5for various porous-effect parameters 11986601 and 11986602 and underthe settings of 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16 Whenthe rotational center is set at (00) as shown in Figure 5 thepresent analytical solutions match well with the publishednumerical results [7] for the limiting case of no porouswall (ie 11986601 = 11986602 = infin) It can be observed thatthe pitching oscillation induced added mass and dampingcoefficients for the floating breakwater system with dualporous side walls are greater than those without porous wallsWith decrease of the porous effect parameters the addedmass and damping coefficients increase For the added masscoefficients it is found that the variations trends are morecomplex for waves with relatively smaller 1198960ℎ under theconditions of different porous-effect parameters Generallyin cases of weaker porous-effect (for example 11986601 = 11986602 le 2as shown in Figure 5(a)) the value of added mass coefficienttends to approach 14 under the case considered For thevariations of damping coefficient as shown in Figure 5(b) thevarying trends are similar to those of damping coefficient insurge motion
The results presented in Figure 6 are the variations ofthe dimensionless surge-pitch added mass and damping
coefficients versus 1198960ℎ for varying 11986601 and 11986602 The case isfor 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16 with referenceto the rotational center of (00) Clearly the added massand damping coefficients are shown to increase with anincrease in the porous-effect parameters of both porouswalls A and B while the damping coefficients reach theirabsolute maximum values when 1198960ℎ is within the rangeof 1 and 2 The magnitude of surge-pitch hydrodynamiccoefficients especially the damping is greatly influenced bythe permeability of the dual porous walls compared to thesituation with surge or pitching motion only
The analytical solutions for the Response AmplitudeOperators (RAOs) of the floating breakwater system pro-posed in this study are illustrated in Figure 7 as a functionof dimensionless wavelength 1198960ℎ for various porous-effectparameters with the conditions of 119904ℎ = 03 119889ℎ = 05119886ℎ = 025 119903119866119886 = 09 and 119866119872119886 = 01 where therotational center is located at (00) It is found that the varyingtrends of surge RAOs are similar to those of pitching RAOsalthough the peak values of surge RAOs are smaller thanthose of pitching RAOs When the porous-effect parametersdecrease the main peak values of either surge or pitchingRAOs that occurred are shifted to the wave conditions withshorter wavelength Furthermore for the heave RAO itexperiences a typical damped response for the case of singledegree of freedom system and its maximum values of RAOwould move to the lower frequency wave region when theporous-effect parameters decrease
5 Summary and Conclusions
The hydrodynamic performance investigations includingthe hydrodynamic loads radiation induced added massand damping coefficients and dynamic responses relatedRAOs are critical to the study that is associated with theproblems of wave and structure interactions In this study the
12 Mathematical Problems in Engineering
G=G=infin
G=G=
G=G=
G=G=
963 100 87521 114kh
minus4
minus3
minus2
minus1
0
1
m
2
as
(a) Added mass
G=G=infin
G=G=
G=G=
G=G=
963 100 87521 114kh
minus3
minus25
minus2
minus15
minus1
minus05
0
d
2
as
(b) Radiation damping
Figure 6 Variation of the dimensionless surge-pitch hydrodynamic coefficients ((a) for the added mass and (b) for the radiation damping)versus 1198960ℎ under the conditions of different porous-effect parameters for 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16 with the rotational center set at(0 0)
G=G=
G=G=
G=G=
G=G=infin
4515 3525 3 605 2 5 551 4kh
0123456789
1011
|X|A
(a) Surge RAO
G=G=infin
G=G=
G=G=
G=G=
3 60 4 51 2kh
0
05
1
15
2
25
3
35|X
|A
(b) Heave RAO
G=G=
G=G=
G=G=
G=G=infin
05 3515 325 45 65 5541 20kh
0
5
10
15
|X|A
(c) Pitch RAO
Figure 7 Variation of the dimensionless RAO ((a) for surge RAO (b) for heave RAO and (c) for pitch RAO) versus 1198960ℎ under the conditionsof different porous-effect parameters for 119904ℎ = 03 119889ℎ = 05 119886ℎ = 025 119903119866119886 = 09 119866119872119886 = 01 with rotation center at (00)
Mathematical Problems in Engineering 13
hydrodynamic problems considering a floating breakwatersystem that consists of a rectangular body and two verticalattached porous walls are investigated analytically based onthe two-dimensional potential flow theory The expressionsof the diffraction and radiation based velocity potentials andthe associated unknown coefficients for each subdomain arederived by the MEEM approach Subsequently the excitinghydrodynamic forces added mass and radiation dampingcoefficients in terms of surge heave and pitching motionsand the equation ofmotion for the floating breakwater systemare formulated and calculated for various case scenariosby varying the wave conditions and porous effect param-eters Considering the limiting case without porous wallsgood agreements between the present analytical solutionsand published numerical results are achieved This suggeststhat the present analytical model is demonstrated to be areliable engineering tool to provide reasonable estimationsfor the hydrodynamic performances of a floating breakwatersystem with dual side porous walls It is worthy to noticethat the wave loads on the floating structures especiallythe moments when referenced to the rotational center ofbottom center would be reduced with an increase in theporous-effect parameters of the porous walls Additionallythe existence of the dual porous walls could move the peaksof surge and pitching RAOs to the waves of higher frequencywhen the porous effect parameters decrease Overall it canbe concluded that a properly adopted porous-wall systemwhen attached to a floating body can effectively improveits hydrodynamic performance as a breakwater The analyt-ical approach developed in this study can also be furtherextended to assist the theoretical analysis for other similar butimproved floating breakwater systems
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was supported by ldquoFundamental Research Fundsfor the Central Universitiesrdquo (Grand No 017180224)
References
[1] I-H Cho ldquoTransmission coefficients of a floating rectangularbreakwater with porous side platesrdquo International Journal ofNaval Architecture and Ocean Engineering vol 8 no 1 pp 53ndash65 2016
[2] H Hu K-H Wang and A N Williams ldquoWave motion over abreakwater system of a horizontal plate and a vertical porouswallrdquo Ocean Engineering vol 29 no 4 pp 373ndash386 2001
[3] C C Mei and J L Black ldquoScattering of surface waves byrectangular obstacles in water of finite depthrdquo Journal of FluidMechanics vol 38 no 3 pp 499ndash511 1969
[4] J L Black C C Mei and M C G Bray ldquoRadiation andscattering of water waves by rigid bodiesrdquo Journal of FluidMechanics vol 46 no 1 pp 151ndash164 1971
[5] N Drimer Y Agnon and M Stiassnie ldquoA simplified analyticalmodel for a floating breakwater in water of finite depthrdquoAppliedOcean Research vol 14 no 1 pp 33ndash41 1992
[6] J-F Lee ldquoOn the heave radiation of a rectangular structurerdquoOcean Engineering vol 22 no 1 pp 19ndash34 1995
[7] Y H Zheng Y G You and Y M Shen ldquoOn the radiationand diffraction of water waves by a rectangular buoyrdquo OceanEngineering vol 31 no 8-9 pp 1063ndash1082 2004
[8] E D Christensen H B Bingham A P Skou Friis A K Larsenand K L Jensen ldquoAn experimental and numerical study offloating breakwatersrdquo Coastal Engineering Journal vol 137 pp43ndash58 2018
[9] E Koutandos P Prinos andXGironella ldquoFloating breakwatersunder regular and irregular wave forcing Reflection and trans-mission characteristicsrdquo Journal of Hydraulic Research vol 43no 2 pp 174ndash188 2005
[10] G H Dong Y N Zheng Y C Li B Teng C T Guan and D FLin ldquoExperiments on wave transmission coefficients of floatingbreakwatersrdquo Ocean Engineering vol 35 no 8-9 pp 931ndash9382008
[11] A T Chwang ldquoA porous-wavemaker theoryrdquo Journal of FluidMechanics vol 132 pp 395ndash406 1983
[12] Y C Li Y Liu and B Teng ldquoPorous effect parameter of thinpermeable platesrdquo Coastal Engineering Journal vol 48 no 4pp 309ndash336 2006
[13] S R Manam and M Sivanesan ldquoScattering of water wavesby vertical porous barriers an analytical approachrdquo WaveMotion An International Journal Reporting Research on WavePhenomena vol 67 pp 89ndash101 2016
[14] A T Chwang and Z Dong ldquoWave-trapping due to a porousplaterdquo in Proceedings of 15 ONR symposium on Naval Hydrody-namics pp 407ndash414 1985
[15] S Zhu and A T Chwang ldquoAnalytical study of porous waveabsorberrdquo Journal of Engineering Mechanics vol 127 no 4 pp326ndash332 2001
[16] H Behera and T Sahoo ldquoHydroelastic analysis of gravity waveinteraction with submerged horizontal flexible porous platerdquoJournal of Fluids and Structures vol 54 pp 643ndash660 2015
[17] T L Yip and A T Chwang ldquoPerforated wall breakwater withinternal horizontal platerdquo Journal of Engineering Mechanics vol126 no 5 pp 533ndash538 2000
[18] Y Liu and H-J Li ldquoWave scattering by dual submergedhorizontal porous plates Further resultsrdquo Ocean Engineeringvol 81 no 5 pp 158ndash163 2014
[19] I H Cho and M H Kim ldquoWave absorbing system usinginclined perforated platesrdquo Journal of Fluid Mechanics vol 608pp 1ndash20 2008
[20] K-H Wang and X Ren ldquoWater waves on flexible and porousbreakwatersrdquo Journal of Engineering Mechanics vol 119 no 5pp 1025ndash1047 1993
[21] K H Wang and X Ren ldquoWave interaction with a concentricporous cylinder systemrdquo Ocean Engineering vol 21 no 4 pp343ndash360 1994
[22] F Zhao T Zhang R Wan L Huang X Wang and WBao ldquoHydrodynamic loads acting on a circular porous platehorizontally submerged in wavesrdquo Ocean Engineering vol 136pp 168ndash177 2017
14 Mathematical Problems in Engineering
[23] A Mondal and R Gayen ldquoWave interaction with dual circularporous platesrdquo Journal of Marine Science and Application vol14 no 4 pp 366ndash375 2015
[24] W L Qiao K-H Wang and Y Q Sun ldquoScattering of waterwaves by a floating body with two vertically attached porouswallsrdquo Journal of Engineering Mechanics vol 144 no 2 ArticleID 04017162 2018
[25] S W Twu and D T Lin ldquoOn a highly effective wave absorberrdquoCoastal Engineering Journal vol 15 no 4 pp 389ndash405 1991
[26] X P Yu ldquoDiffraction of water waves by porous breakwatersrdquoJournal of Waterway Port Coastal and Ocean Engineering vol121 no 6 pp 275ndash282 1995
[27] I J Losada M A Losada and A Baquerizo ldquoAn analyticalmethod to evaluate the efficiency of porous screens as wavedampersrdquo Applied Ocean Research vol 15 no 4 pp 207ndash2151993
[28] G Besio and M A Losada ldquoSediment transport patterns atTrafalgar offshore windfarmrdquo Ocean Engineering vol 35 no 7pp 653ndash665 2008
[29] H Song and L Tao ldquoShort-crestedwave interactionwith a con-centric porous cylindrical structurerdquo Applied Ocean Researchvol 29 no 4 pp 199ndash209 2007
[30] L Xiao Y Kou L Tao and L Yang ldquoComparative studyof hydrodynamic performances of breakwaters with double-layered perforated walls attached to ring-shaped very largefloating structuresrdquo Ocean Engineering vol 111 pp 279ndash2912016
[31] T L Yip and A T Chwang ldquoWater wave control by submergedpitching porous platerdquo Journal of Engineering Mechanics vol124 no 4 pp 428ndash434 1998
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Mathematical Problems in Engineering 5
23 Analytical Solutions of the Radiated Velocity PotentialsSimilar to the derivation of the solutions of the diffractedvelocity potentials the application of the method of sepa-ration variables to the governing equation (22) under theboundary conditions (23) (24) and (25) yields the radiatedvelocity potentials as follows
120601(119871)1198771 = infinsum119898=0
119860(119871)119898 119890minus119896119898(119909+119886) cos 119898 (119911 + ℎ)cos 119898ℎ 119890minusi120596119905 (26)
120601(1)1198772 = (119862(1)0 119909 + 119864(1)0 ) 119890minusi120596119905+ infinsum119898=1
[119862(1)119898 cosh (119903119898119909)cosh (119903119886119909) + 119864(1)119898 sinh (119903119898119909)
sinh (119903119886119909) ]sdot cos 119903119898 (119911 + ℎ) 119890minusi120596119905
(27)
120601(2)1198772 = (119911 + ℎ)2 minus 11990922 (ℎ minus 119904) + (119862(2)0 119909 + 119864(2)0 ) 119890minusi120596119905
+ infinsum119898=1
[119862(2)119898 cosh (119903119898119909)cosh (119903119886119909) + 119864(2)119898 sinh (119903119898119909)
sinh (119903119886119909) ]sdot cos 119903119898 (119911 + ℎ) 119890minusi120596119905
(28)
120601(3)1198772 = infinsum119898=1
120576 cos [120574119898 (119909 minus 119886)] cosh [120574119898 (119911 + ℎ)]+ (119862(3)0 119909 + 119864(3)0 ) 119890minusi120596119905+ infinsum119898=1
[119862(3)119898 cosh (119903119898119909)cosh (119903119886119909) + 119864(3)119898 sinh (119903119898119909)
sinh (119903119886119909) ]sdot cos 119903119898 (119911 + ℎ) 119890minusi120596119905
(29)
120601(119871)1198773 = infinsum119898=0
119861(119871)119898 119890119896119898(119909minus119886) cos 119898 (119911 + ℎ)cos 119898ℎ 119890minusi120596119905 (30)
where
120574119898 = 1198981205872119886 (119898 = 1 2 3 ) (31)
Combining the floating body surface condition at 119911 = minus119904minus119886 le 119909 le 119886 and the orthogonality of cos[120574119898(119909 minus 119886)] over theintegral interval of minus119886 le 119909 le 119886 the expression of 120576 can bederived as
120576 = cos 2119886120574119898sinh [120574119898 (ℎ minus 119904)] 1198861205741198983 (32)
24 Determination of theUnknownCoefficientsatAppearedin the Solutions of Radiated Velocity Potentials The solutionsof the unknown coefficients that appeared in the radiationpotentials can be again obtained by utilizing the matchingconditions at 119909 = plusmn119886 under the principle of continuouspressure and normal velocity We have the following
(a) Matching Conditions for the Surge-Induced Potential
120601(1)1198772 = 120601(1)1198771 minusℎ le 119911 le minus119889 119909 = minus119886120601(1)1198773 minusℎ le 119911 le minus119889 119909 = 119886 (33a)
120597120601(1)1198771120597119909 = 120597120601(1)1198772120597119909 minusℎ le 119911 le minus119889 119909 = minus1198861 minus119904 le 119911 le 0 119909 = 119886 (33b)
120597120601(1)1198773120597119909 = 120597120601(1)1198772120597119909 minusℎ le 119911 le minus119889 119909 = minus1198861 minus119904 le 119911 le 0 119909 = 119886 (33c)
(b) Matching Conditions for the Heave-Induced Potential
120601(2)1198772 = 120601(2)1198771 minusℎ le 119911 le minus119889 119909 = minus119886120601(2)1198773 minusℎ le 119911 le minus119889 119909 = 119886 (34a)
120597120601(2)1198771120597119909 = 120597120601(2)1198772120597119909 minusℎ le 119911 le minus119889 119909 = minus1198861 minus119904 le 119911 le 0 119909 = 119886 (34b)
120597120601(2)1198773120597119909 = 120597120601(2)1198772120597119909 minusℎ le 119911 le minus119889 119909 = minus1198861 minus119904 le 119911 le 0 119909 = 119886 (34c)
(c) Matching Conditions for the Pitching-Induced Potential
120601(3)1198772 = 120601(3)1198771 minusℎ le 119911 le minus119889 119909 = minus119886120601(3)1198773 minusℎ le 119911 le minus119889 119909 = 119886 (35a)
120597120601(3)1198771120597119909 = 120597120601(3)1198772120597119909 minusℎ le 119911 le minus119889 119909 = minus119886119911 minus 120585119911 minus119904 le 119911 le 0 119909 = 119886 (35b)
120597120601(3)1198773120597119909 = 120597120601(3)1198772120597119909 minusℎ le 119911 le minus119889 119909 = minus119886119911 minus 120585119911 minus119904 le 119911 le 0 119909 = 119886 (35c)
For the porous walls A and B at the boundaries 119909 =plusmn119886 the condition of continuity of normal velocity for flowspassing through the porous walls A and B leads to
120597120601(119871)1198771120597119909 = 120597120601(119871)1198772120597119909 = 119882119860 (119911 119905)119871 = (1 2 3) minus ℎ le 119911 le minus119889 119909 = minus119886
(36a)
120597120601(119871)1198773120597119909 = 120597120601(119871)1198772120597119909 = 119882119861 (119911 119905)119871 = (1 2 3) minus ℎ le 119911 le minus119889 119909 = minus119886
(36b)
where the porous flow velocities 119882119860(119911 119905) and 119882119861(119911 119905) areassumed to obey Darcys law that is they are linearly pro-portional to the pressure difference induced by the radiated
6 Mathematical Problems in Engineering
motions across their corresponding porous wall (Chwang[11]) With the application of the linearized Bernoulli equa-tion we have
119882119860 (119911 119905) = minus11986601 [ 1198960120596 (120597120601(119871)1198771120597119905 minus 120597120601(119871)1198772120597119905 )](119871 = 1 2 3) 119909 = minus119886 minus119889 le 119911 le minus119904
(37a)
119882119861 (119911 119905) = minus11986602 [ 1198960120596 (120597120601(119871)1198772120597119905 minus 120597120601(119871)1198773120597119905 )](119871 = 1 2 3) 119909 = 119886 minus119889 le 119911 le minus119904
(37b)
Utilizing the orthogonality property of eigenfunctionscos 119899(119911 + ℎ) for 120601(119871)1198771 and 120601(119871)1198773 (119871 = 1 2 3) over the integralinterval of minusℎ le 119911 le 0 the following equations can beformulated by the matching conditions of (33b) (33c) (34b)(34c) (35b) and (35c) as
int0minusℎ
120597120601(1)1198771120597119909 cos 119899 (119911 + ℎ) 119889119911= intminus119904minusℎ
120597120601(1)1198772120597119909 cos 119899 (119911 + ℎ) 119889119911+ int0minus119904
1 cos 119899 (119911 + ℎ) 119889119911 (119909 = minus119886)(38)
int0minusℎ
120597120601(1)1198773120597119909 cos 119899 (119911 + ℎ) 119889119911= intminus119904minusℎ
120597120601(1)1198772120597119909 cos 119899 (119911 + ℎ) 119889119911+ int0minus119904
1 cos 119899 (119911 + ℎ) 119889119911 (119909 = 119886)(39)
int0minusℎ
120597120601(2)1198771120597119909 cos 119899 (119911 + ℎ) 119889119911= intminus119904minusℎ
120597120601(2)1198772120597119909 cos 119899 (119911 + ℎ) 119889119911 (119909 = minus119886)(40)
int0minusℎ
120597120601(2)1198773120597119909 cos 119899 (119911 + ℎ) 119889119911= intminus119904minusℎ
120597120601(2)1198772120597119909 cos 119899 (119911 + ℎ) 119889119911 (119909 = 119886)(41)
int0minusℎ
120597120601(3)1198771120597119909 cos 119899 (119911 + ℎ) 119889119911= intminus119904minusℎ
120597120601(3)1198772120597119909 cos 119899 (119911 + ℎ) 119889119911+ int0minus119904
(119911 minus 120585119911) cos 119899 (119911 + ℎ) 119889119911 (119909 = minus119886)(42)
int0minusℎ
120597120601(3)1198773120597119909 cos 119899 (119911 + ℎ) 119889119911
= intminus119904minusℎ
120597120601(3)1198772120597119909 cos 119899 (119911 + ℎ) 119889119911+ int0minus119904
(119911 minus 120585119911) cos 119899 (119911 + ℎ) 119889119911 (119909 = 119886)
(43)
Additionally by combining (36a) (36b) (37a) and (37b)with the matching conditions given in (33a) (34a) and(35a) respectively and applying the orthogonality propertyof eigenfunctions cos 119903119899(119911 + ℎ) for 120601(119871)1198772 (L = 1 2 3) over theintegral interval of minusℎ le 119911 le minus119904 we have
intminus119904minusℎ
120601(119871)1198772 cos 119903119899 (119911 + ℎ) 119889119911= intminus119889minusℎ
120601(119871)1198771 cos 119903119899 (119911 + ℎ) 119889119911+ intminus119904minus119889
(119871)1198771 cos 119903119899 (119911 + ℎ) 119889119911(119909 = minus119886 119871 = 1 2 3)
(44)
intminus119904minusℎ
120601(119871)1198772 cos 119903119899 (119911 + ℎ) 119889119911= intminus119889minusℎ
120601(119871)1198773 cos 119903119899 (119911 + ℎ) 119889119911+ intminus119904minus119889
(119871)1198773 cos 119903119899 (119911 + ℎ) 119889119911(119909 = 119886 119871 = 1 2 3)
(45)
where
(1)1198771 = infinsum119898=0
119860(1)119898 (1 + 119898i119866011198960)
cos 119898 (119911 + ℎ)cos 119898ℎ
+ 1i119866011198960
(46)
(2)1198771 = infinsum119898=0
119860(2)119898 (1 + 119898i119866011198960)
cos 119898 (119911 + ℎ)cos 119898ℎ (47)
(3)1198771 = infinsum119898=0
119860(3)119898 (1 + 119898i119866011198960)
cos 119898 (119911 + ℎ)cos 119898ℎ
+ 119911i119866011198960
(48)
Mathematical Problems in Engineering 7
(1)1198773 = infinsum119898=0
119861(1)119898 (1 + 119898i119866021198960)
cos 119898 (119911 + ℎ)cos 119898ℎ
minus 1i119866021198960
(49)
(2)1198773 = infinsum119898=0
119861(2)119898 (1 minus 119898i119866021198960)
cos 119898 (119911 + ℎ)cos 119898ℎ (50)
(3)1198773 = infinsum119898=0
119861(3)119898 (1 + 119898i119866021198960)
cos 119898 (119911 + ℎ)cos 119898ℎ
minus 119911i119866021198960
(51)
Again the summation of terms with the infinite seriesis approximated by a large but finite numbers of termseg a total of M terms Then the unknown coefficients119860(1)119898 119862(1)119898 119864(1)119898 119861(1)119898 (119898 = 0 1 2 119872) [4(M+1) unknownstotally] can be determined by solving the 4(M+1) sys-tem of algebraic equations formulated in (38) (39) (44)and (45) (for L=1) Similarly the unknown coefficients119860(2)119898 119862(2)119898 119864(2)119898 119861(2)119898 (119898 = 0 1 2 119872) can be calculatedfrom the 4(M+1) system of algebraic equations expressedin (40) (41) (44) and (45) (for L=2) while the coefficients119860(3)119898 119862(3)119898 119864(3)119898 119861(3)119898 (119898 = 0 1 2 119872) can be evaluated bysolving the 4(M+1) system of algebraic equations formulatedin (42) (43) (44) and (45) (for L=3) After the determinationof the above unknown coefficients a series of engineeringproperties important to practical applications could be cal-culated for example the hydrodynamic coefficients and themotion induced dynamic responses of the breakwater system
3 Exciting Forces andHydrodynamic Coefficients
31 Exciting Forces To determine the hydrodynamic coeffi-cients related to the inducedmotions of the floating breakwa-ter system the exciting forces and moment of the combinedincident and diffracted waves on the breakwater need tobe computed as inputs for solving the equation of motionBy integrating the dynamic pressures with the use of thevelocity potentials of incident and diffracted waves (see (7)-(9)) over the wet surface the hydrodynamic forces alongthe x direction (1198651198631) and along the z direction (1198651198632) can beformulated respectively as (Qiao et al [24])
1198651198631= i120588120596 infinsum
119898=0
(119868119898 + 119865119898 minus 119879119898) [sin 119898ℎ minus sin 119898 (ℎ minus 119889)]119898 cos 119898ℎ
sdot 119890minusi120596119905 + i120588120596 lowast 2119886 (119889 minus 119904) 1198620119890minusi120596119905minus i120588120596 infinsum
119898=0
2119864119898119903119898 sin 119903119898 (ℎ minus 119889) 119890minusi120596119905(52)
1198651198632 = i120596120588 [21198861198640 + 2 infinsum119898=1
119862119898119903119898 tanh (119903119898119886) cos 119903119898 (ℎ minus 119904)]sdot 119890minusi120596119905
(53)
Also the wave exciting moment (1198651198633) that is referencedto a rotational center (0120585119911) can be derived as
1198651198633 = i120588120596
(119868119898 + 119865119898 minus 119879119898)
sdot [[[[[[
119889 sin 119898 (ℎ minus 119889) minus 120585119911 sin 119898ℎ + 120585119911 sin 119898 (ℎ minus 119889)119898 cos 119898ℎ
+cos 119898ℎ minus cos 119898 (ℎ minus 119889)1198982 cos 119898ℎ
]]]]]]
sdot 119890minusi120596119905 + i120588120596 1198861198620 (1198892 minus 1199042)
minus infinsum119898=1
2119864119898119903119898 [119889 sin 119903119898 (ℎ minus 119889)
+ cos 119903119898 (ℎ minus 119904) minus cos 119903119898 (ℎ minus 119889)119903119898 ] 119890minusi120596119905
+ i120588120596 [21198861198620120585119911 (119889 minus 119904) minus infinsum119898=1
2119864119898120585119911119903119898 sin 119903119898 (ℎ minus 119889)]
sdot 119890minusi120596119905 minus i120588120596 2119862011988633 + infinsum119898=1
[ 2119886119864119898 cos 119903119898 (ℎ minus 119904)119903119898 tanh (119903119898119886)minus 2119864119898 cos 119903119898 (ℎ minus 119904)1199031198982 ] 119890minusi120596119905
(54)
32 Hydrodynamic Coefficients The radiation forces inducedby the translational and rotational motions of the floatingbreakwater system can be calculated from the derived radi-ation potentials utilizing the linearized Bernoulli equation as
119865119871119896 = i120588120596 intΓ
Φ(119871)119877119896 119890minusi120596119905119899119895119889Γ= 12058812059621198830119895 int
Γ120601(119871)119877119896 119890minusi120596119905119899119895119889Γ
= 12059621198830119895119898119871119896119890minusi120596119905 + i1205961198830119895119889119871119896119890minusi120596119905(55)
where 119898119871119896 is the radiation-induced added mass and the119889119871119896 denotes the radiation damping The expressions arerespectively
119898119871119896 = Re(120588 intΓ
120601(119871)119877119896119899119895119889Γ) = Re (120588119871119896) (56)
119889119871119896 = Im (120588120596 intΓ
120601(119871)119877119896119899119895119889Γ) = Im (120588120596119871119896) (57)
8 Mathematical Problems in Engineering
119871119896 can be computed with formulations given below
11 = (int0minus119889
120601(1)1198771 10038161003816100381610038161003816119909=minus119886 119889119911 minus int0minus119889
120601(1)1198773 10038161003816100381610038161003816119909=119886 119889119911)+ (intminus119904minus119889
120601(1)1198772 10038161003816100381610038161003816119909=minus119886 119889119911 minus intminus119904minus119889
120601(1)1198772 10038161003816100381610038161003816119909=119886 119889119911)= 119872sum119898=1
119860(1)119898 minus 119861(1)119898119898 cos 119898ℎ [sin 119898ℎ minus sin 119898 (ℎ minus 119889)]+ 2119886119862(1)0 (119889 minus 119904) + 119872sum
119898=1
2119864(1)119898119903119898 [sin119898120587minus sin 119903119898 (ℎ minus 119889)]
(58)
22 = int119886minus119886
120601(2)1198772 10038161003816100381610038161003816119911=minus119909 119889119909 = 2119886119864(2)0+ 119872sum119898=1
2119862(2)119898119903119898 tanh (119903119898119886) cos 119903119898 (ℎ minus 119904) + 119886 (ℎ minus 119904)minus 11988633 (ℎ minus 119904)
(59)
33 = int0minus119889
( 120601(3)1198771 10038161003816100381610038161003816119909=minus119886 minus 120601(3)1198773 10038161003816100381610038161003816119909=119886) (119911 minus 120585119911) 119889119911+ intminus119904minus119889
(120601(3)1198772 10038161003816100381610038161003816119909=minus119886 minus 120601(3)1198772 10038161003816100381610038161003816119909=119886) (119911 minus 120585119911) 119889119911minus int119886minus119886
120601(3)1198772 10038161003816100381610038161003816119909=minus119904 119909 119889119909 = 119872sum119898=0
(119860(3)119898 minus 119861(3)119898 )
sdot [119889 sin 119898 (ℎ minus 119889) minus 120585119911 sin 119898 (ℎ minus 119889)119898 cos 119898ℎ
+ cos 119898ℎ minus cos 119898 (ℎ minus 119889)1198982 cos 119898ℎ ] + 119886119862(3)0 (1198892 minus 1199042)
minus 119872sum119898=1
2119864(3)119898119903119898 [119889 sin 119898 (ℎ minus 119889)+ cos 119903119898 (ℎ minus 119904) minus cos 119903119898 (ℎ minus 119889)119903119898 ] + 2119886119862(3)0 120585119911 (119889minus 119904) minus 119872sum
119898=1
2119864(3)119898 120585119911119903119898 sin 119903119898 (ℎ minus 119889) minus 23 119862(3)0 1198863 minus 119872sum119898=1
120576sdot cosh 120574119898 (ℎ minus 119904) 1 minus cos 21198861205741198981205741198982 + 119872sum
119898=1
(120576 cos119898120587minus 120576) [ 119889 sinh 120574119898 (ℎ minus 119889) minus 119904 sinh 120574119898 (ℎ minus 119904)120574119898minus cosh 120574119898 (ℎ minus 119904) minus cosh 120574119898 (ℎ minus 119889)1205741198982 ]minus 119872sum119898=1
119864(3)119898 cos 119903119898 (h minus s)119903119898 sinh 119903119898119886 (2119886 cosh 119903119898119886minus 2 sinh 119903119898119886119903119898 )
(60)
31 = 13 = int0minus119889
( 120601(3)1198771 10038161003816100381610038161003816119909=minus119886 minus 120601(3)1198773 10038161003816100381610038161003816119909=119886) 119889119911+ intminus119904minus119889
( 120601(3)1198772 10038161003816100381610038161003816119909=minus119886 minus 120601(3)1198772 10038161003816100381610038161003816119909=119886) 119889119911 = 119872sum119898=0
(119860(3)119898minus 119861(3)119898 ) sin 119898ℎ minus sin 119898 (ℎ minus 119889)
119898 cos 119898ℎ minus 2 (ℎ minus 119904)sdot [ 11988636 (ℎ minus 119904) + 119886119862(3)0 ] + 119886 (ℎ minus 119904)3 minus 119886 (ℎ minus 119889)33 (ℎ minus 119904)+ 119872sum119898=1
2119864(3)119898119903119898 sin 119903119898 (ℎ minus 119889)
(61)
33 Equation of Motion and RAOs The derived solutions forthe diffraction and radiation problems could now be inte-grated with the equation of motion of the floating breakwatersystem to analyze the dynamic responses of the structuralsystem We have
minus1205962 [119872 + 119898] minus i120596 [119889] + [119904] [1198830] = [1198650] (62)
where [1198830] = [11988301 11988302 11988303]119879 and are vectors indicatingthe motion amplitudes and the wave exciting forces respec-tively [119898] and [119889] are addedmass and dampingmatrices [119872]is the mass matrix and [119878] denotes the hydrostatic restoringforce matrix they can be expressed as
[119872] = [[[
1198980 0 minus11989801198851198660 1198980 0minus1198980119885119866 0 1]]]
(63)
[119878] = [[[
0 0 00 2120588119892119886 00 0 1198980119892119866119872
]]]
(64)
Here 1198980 is the mass per unit length of the floatingbreakwater and 119911119866 is the z-coordinate of the center of gravityThe mass moment of inertia about y-axis I can be computedby 119868 = 11989801199031198662 in which 119903119866 is the radius of gyration in pitchingoscillation and 119866119872 is the metacentric height of the floatingbreakwater system Once 1198830119895 (j=1 2 3) are determined bysolving (62) the RAOs (Response Amplitude Operators) ofthe breakwater system can be calculated as
119877119860119874119895 = 1198830119895119867119895 (119895 = 1 2 3) (65)
4 Results and Discussion
The validation of the wave diffraction solutions in terms ofreflectiontransmission coefficients and hydrodynamic loadsfor a train of monochromatic wave propagating throughthe proposed floating breakwater system that consists of apartially submerged rectangular body and with or withoutthe attached two porous side walls has been carried outin a previous study (Qiao et al [24]) Here additional
Mathematical Problems in Engineering 9
Black et al(1971)Present approach
G=G=infinhs=as=
0010203040506070809
F D
gH
a
3 60 5 71 42ks
(a) Horizontal force
Black et al(1971)Present approach
G=G=infinhs= as=
0010203040506070809
1
F D
gHa
3 60 5 71 42ks
(b) Vertical force
Black et al(1971)Present approach
G=G=infinhs= as=
3 60 4 51 72ks
0
005
01
015
02
025
F D
gH
a2
(c) Moment
Figure 2 Comparison of the dimensionless wave exciting forces and moment versus 1198960119904 for ℎ119904 = 20 119886119904 = 30 11986601 = 11986602 997888rarr infin (a)dimensionless horizontal exciting force (b) dimensionless vertical exciting force (c) dimensionless exciting moment about the rotationalcenter of (0 0)
comparisons of wave diffraction results with other publishednumerical solutions for the limiting case of neglecting theeffect of two porous walls are again presented For thewave radiation problem the motion induced hydrodynamiccoefficients such as added mass and damping coefficient arealso investigated for their variations with changes of waveparameters As a validation those results for the caseswithoutporous walls are compared with limited numerical solutionspublished in literatures Subsequently the effects of variousdimensionless parameters for example porous propertiesincident wave conditions and the geometry of floating bodyto wave-induced hydrodynamic forces hydrodynamic coeffi-cients and RAOs were also examined In our computationsthe first 40 terms in the infinite series of the diffractionpotentials and radiation potentials are taken the same as [7]namely M=40
As an example showing the results comparisons for thediffraction problem Figure 2 illustrates the variations ofthe dimensionless exciting forces in horizontal and verticaldirections versus dimensionless wavelength 1198960119904 for the caseof ℎ119904 = 20 119886119904 = 10 11986601 = 11986602 997888rarr infin The excitingmoments with reference to the rotational center of (00) are
also presented in Figure 2 Here 11986601 = 11986602 997888rarr infin representsthe case of a floating breakwater system without dual porousside walls The numerical results computed by Black et al [4]are also plotted in Figure 2 The comparisons indicate thatthe exciting forces and moments are well predicted by thepresent analytical model Both the analytical solutions andthe numerical results reveal that the horizontal exciting forcesand the exciting moments show rapidly increasing trendinitially until a peak value is reached and then are followedby a gradual decrease of the values with the further increaseof 1198960119904 However the vertical forces as shown in Figure 2(b)decrease continuously with an increase in 1198960119904 According tothe results given in the previous study by Qiao et al [24] itindicated that the effectiveness of the porous walls is reflectedwith the reduction of the vertical hydrodynamic forces andwhen the breakwater system is rotated with reference to thebottom center of the body the induced moment can also bereduced However due to the added additional length of theporous walls the horizontal forces are slightly greater thanthose without porous walls
Figure 3 presents the variations of the dimensionlessadded mass and damping coefficients in surge only motion
10 Mathematical Problems in Engineering
YHZheng et al(2004)
G=G=
G=G=
G=G=
G=G=infin
0
1
2
3
4
5
6m
2
as
963 100 87521 114kh
(a) Added mass
YH Zheng et al(2004)
G=G=infin
G=G=
G=G=
G=G=
963 100 87521 114kh
005
115
225
335
4
d
2
as
(b) Radiation coefficient
Figure 3 Variation of the dimensionless surge-surge hydrodynamic coefficients ((a) for the added mass and (b) for the radiation damping)versus 1198960ℎ under the conditions of different porous-effect parameters for 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16
YHZheng et al(2004)
G=G=
G=G=
G=G=
G=G=infin
963 100 87521 114kh
0
02
04
06
08
1
12
14
m
2
as
(a) Added mass
YHZheng et al(2004)
G=G=
G=G=
G=G=
G=G=infin
minus04minus02
002040608
11214
d
2
as
963 100 87521 114kh
(b) Radiation damping
Figure 4 Variation of the dimensionless heave-heave hydrodynamic coefficients ((a) for the added mass and (b) for the radiation damping)versus 1198960ℎ under the conditions of different porous-effect parameters for 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16
versus 1198960ℎ for the setting of 119904ℎ = 13 119889ℎ = 12 and 119886ℎ =16under the conditions of various porous-effect parametersWhen 11986601 = 11986602 = infin it represents the case withoutconsidering the effect of dual porous walls The presentanalytical solutions for this limiting case can be adopted tocompare to the limited numerical results available in thestudy of [7] The comparison plot is shown in Figure 4 wherethe nearly identical matches between the present analyticalsolutions and the numerical results of [7] can be observedThe results shown in Figure 4 suggest that the addition ofporous walls tends to enhance the added mass and dampingcoefficients In general with a decrease of porous effectparameter the addedmass and damping coefficients increaseFor the variation of addedmass coefficient with porouswalls
it can be seen that a peak value occurred at a long wave (small1198960ℎ value) condition However for the radiation dampingcoefficient it is shown to increase rapidly to its maximumvalue with an increase of 1198960ℎ then a gradual decrease trendis followed by a further increase of 1198960ℎ
The dimensionless heave-heave added mass (119898222120588119886119904)and radiation damping coefficient (119889222120588120596119886119904) versus 1198960ℎunder various 11986601 and 11986602 values for the case of 119904ℎ =13 119889ℎ = 12 119886ℎ = 16 are presented in Figure 4Under the limiting condition of 11986601 = 11986602 = infin (withoutporous walls) the numerical results produced by the study of[7] are also plotted for comparison Again good agreementsbetween the present analytical solutions and the publishednumerical results can be noticed The results in Figure 4(a)
Mathematical Problems in Engineering 11
YHZheng et al(2004)
963 100 87521 114kh
0
02
04
06
08
1
12
14m
2
as
G=G=infin
G=G=
G=G=
G=G=
(a) Added mass
YH Zhang et al(2004)
G=G=infin
G=G=
G=G=
G=G=
0010203040506070809
d
2
as
963 100 87521 114kh
(b) Radiation damping
Figure 5 Variation of the dimensionless pitch-pitch hydrodynamic coefficients ((a) for the added mass and (b) for the radiation damping)versus 1198960ℎ under the conditions of different porous-effect parameters for 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16 with the rotational center set at (00)
suggest that the variations of added mass for the shorterwavelength waves (large 1198960ℎ values) are less sensitive to thechange of porous parameters The presence of dual porouswalls increases the heave-heave added mass which tends toreach its maximum value at the long wave extreme Alsoa decrease in porous parameters results in an increase inheave induced added mass As shown in Figure 4(b) theheave-heave damping coefficients decrease rapidly when 1198960ℎincreases initially Then a gradual changing trend is followedby a further increase in 1198960ℎ Additionally the presence ofdual porous walls tends to reduce the values of heave-heavedamping coefficients
For the pitching induced added mass and dampingcoefficients the results are plotted against 1198960ℎ in Figure 5for various porous-effect parameters 11986601 and 11986602 and underthe settings of 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16 Whenthe rotational center is set at (00) as shown in Figure 5 thepresent analytical solutions match well with the publishednumerical results [7] for the limiting case of no porouswall (ie 11986601 = 11986602 = infin) It can be observed thatthe pitching oscillation induced added mass and dampingcoefficients for the floating breakwater system with dualporous side walls are greater than those without porous wallsWith decrease of the porous effect parameters the addedmass and damping coefficients increase For the added masscoefficients it is found that the variations trends are morecomplex for waves with relatively smaller 1198960ℎ under theconditions of different porous-effect parameters Generallyin cases of weaker porous-effect (for example 11986601 = 11986602 le 2as shown in Figure 5(a)) the value of added mass coefficienttends to approach 14 under the case considered For thevariations of damping coefficient as shown in Figure 5(b) thevarying trends are similar to those of damping coefficient insurge motion
The results presented in Figure 6 are the variations ofthe dimensionless surge-pitch added mass and damping
coefficients versus 1198960ℎ for varying 11986601 and 11986602 The case isfor 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16 with referenceto the rotational center of (00) Clearly the added massand damping coefficients are shown to increase with anincrease in the porous-effect parameters of both porouswalls A and B while the damping coefficients reach theirabsolute maximum values when 1198960ℎ is within the rangeof 1 and 2 The magnitude of surge-pitch hydrodynamiccoefficients especially the damping is greatly influenced bythe permeability of the dual porous walls compared to thesituation with surge or pitching motion only
The analytical solutions for the Response AmplitudeOperators (RAOs) of the floating breakwater system pro-posed in this study are illustrated in Figure 7 as a functionof dimensionless wavelength 1198960ℎ for various porous-effectparameters with the conditions of 119904ℎ = 03 119889ℎ = 05119886ℎ = 025 119903119866119886 = 09 and 119866119872119886 = 01 where therotational center is located at (00) It is found that the varyingtrends of surge RAOs are similar to those of pitching RAOsalthough the peak values of surge RAOs are smaller thanthose of pitching RAOs When the porous-effect parametersdecrease the main peak values of either surge or pitchingRAOs that occurred are shifted to the wave conditions withshorter wavelength Furthermore for the heave RAO itexperiences a typical damped response for the case of singledegree of freedom system and its maximum values of RAOwould move to the lower frequency wave region when theporous-effect parameters decrease
5 Summary and Conclusions
The hydrodynamic performance investigations includingthe hydrodynamic loads radiation induced added massand damping coefficients and dynamic responses relatedRAOs are critical to the study that is associated with theproblems of wave and structure interactions In this study the
12 Mathematical Problems in Engineering
G=G=infin
G=G=
G=G=
G=G=
963 100 87521 114kh
minus4
minus3
minus2
minus1
0
1
m
2
as
(a) Added mass
G=G=infin
G=G=
G=G=
G=G=
963 100 87521 114kh
minus3
minus25
minus2
minus15
minus1
minus05
0
d
2
as
(b) Radiation damping
Figure 6 Variation of the dimensionless surge-pitch hydrodynamic coefficients ((a) for the added mass and (b) for the radiation damping)versus 1198960ℎ under the conditions of different porous-effect parameters for 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16 with the rotational center set at(0 0)
G=G=
G=G=
G=G=
G=G=infin
4515 3525 3 605 2 5 551 4kh
0123456789
1011
|X|A
(a) Surge RAO
G=G=infin
G=G=
G=G=
G=G=
3 60 4 51 2kh
0
05
1
15
2
25
3
35|X
|A
(b) Heave RAO
G=G=
G=G=
G=G=
G=G=infin
05 3515 325 45 65 5541 20kh
0
5
10
15
|X|A
(c) Pitch RAO
Figure 7 Variation of the dimensionless RAO ((a) for surge RAO (b) for heave RAO and (c) for pitch RAO) versus 1198960ℎ under the conditionsof different porous-effect parameters for 119904ℎ = 03 119889ℎ = 05 119886ℎ = 025 119903119866119886 = 09 119866119872119886 = 01 with rotation center at (00)
Mathematical Problems in Engineering 13
hydrodynamic problems considering a floating breakwatersystem that consists of a rectangular body and two verticalattached porous walls are investigated analytically based onthe two-dimensional potential flow theory The expressionsof the diffraction and radiation based velocity potentials andthe associated unknown coefficients for each subdomain arederived by the MEEM approach Subsequently the excitinghydrodynamic forces added mass and radiation dampingcoefficients in terms of surge heave and pitching motionsand the equation ofmotion for the floating breakwater systemare formulated and calculated for various case scenariosby varying the wave conditions and porous effect param-eters Considering the limiting case without porous wallsgood agreements between the present analytical solutionsand published numerical results are achieved This suggeststhat the present analytical model is demonstrated to be areliable engineering tool to provide reasonable estimationsfor the hydrodynamic performances of a floating breakwatersystem with dual side porous walls It is worthy to noticethat the wave loads on the floating structures especiallythe moments when referenced to the rotational center ofbottom center would be reduced with an increase in theporous-effect parameters of the porous walls Additionallythe existence of the dual porous walls could move the peaksof surge and pitching RAOs to the waves of higher frequencywhen the porous effect parameters decrease Overall it canbe concluded that a properly adopted porous-wall systemwhen attached to a floating body can effectively improveits hydrodynamic performance as a breakwater The analyt-ical approach developed in this study can also be furtherextended to assist the theoretical analysis for other similar butimproved floating breakwater systems
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was supported by ldquoFundamental Research Fundsfor the Central Universitiesrdquo (Grand No 017180224)
References
[1] I-H Cho ldquoTransmission coefficients of a floating rectangularbreakwater with porous side platesrdquo International Journal ofNaval Architecture and Ocean Engineering vol 8 no 1 pp 53ndash65 2016
[2] H Hu K-H Wang and A N Williams ldquoWave motion over abreakwater system of a horizontal plate and a vertical porouswallrdquo Ocean Engineering vol 29 no 4 pp 373ndash386 2001
[3] C C Mei and J L Black ldquoScattering of surface waves byrectangular obstacles in water of finite depthrdquo Journal of FluidMechanics vol 38 no 3 pp 499ndash511 1969
[4] J L Black C C Mei and M C G Bray ldquoRadiation andscattering of water waves by rigid bodiesrdquo Journal of FluidMechanics vol 46 no 1 pp 151ndash164 1971
[5] N Drimer Y Agnon and M Stiassnie ldquoA simplified analyticalmodel for a floating breakwater in water of finite depthrdquoAppliedOcean Research vol 14 no 1 pp 33ndash41 1992
[6] J-F Lee ldquoOn the heave radiation of a rectangular structurerdquoOcean Engineering vol 22 no 1 pp 19ndash34 1995
[7] Y H Zheng Y G You and Y M Shen ldquoOn the radiationand diffraction of water waves by a rectangular buoyrdquo OceanEngineering vol 31 no 8-9 pp 1063ndash1082 2004
[8] E D Christensen H B Bingham A P Skou Friis A K Larsenand K L Jensen ldquoAn experimental and numerical study offloating breakwatersrdquo Coastal Engineering Journal vol 137 pp43ndash58 2018
[9] E Koutandos P Prinos andXGironella ldquoFloating breakwatersunder regular and irregular wave forcing Reflection and trans-mission characteristicsrdquo Journal of Hydraulic Research vol 43no 2 pp 174ndash188 2005
[10] G H Dong Y N Zheng Y C Li B Teng C T Guan and D FLin ldquoExperiments on wave transmission coefficients of floatingbreakwatersrdquo Ocean Engineering vol 35 no 8-9 pp 931ndash9382008
[11] A T Chwang ldquoA porous-wavemaker theoryrdquo Journal of FluidMechanics vol 132 pp 395ndash406 1983
[12] Y C Li Y Liu and B Teng ldquoPorous effect parameter of thinpermeable platesrdquo Coastal Engineering Journal vol 48 no 4pp 309ndash336 2006
[13] S R Manam and M Sivanesan ldquoScattering of water wavesby vertical porous barriers an analytical approachrdquo WaveMotion An International Journal Reporting Research on WavePhenomena vol 67 pp 89ndash101 2016
[14] A T Chwang and Z Dong ldquoWave-trapping due to a porousplaterdquo in Proceedings of 15 ONR symposium on Naval Hydrody-namics pp 407ndash414 1985
[15] S Zhu and A T Chwang ldquoAnalytical study of porous waveabsorberrdquo Journal of Engineering Mechanics vol 127 no 4 pp326ndash332 2001
[16] H Behera and T Sahoo ldquoHydroelastic analysis of gravity waveinteraction with submerged horizontal flexible porous platerdquoJournal of Fluids and Structures vol 54 pp 643ndash660 2015
[17] T L Yip and A T Chwang ldquoPerforated wall breakwater withinternal horizontal platerdquo Journal of Engineering Mechanics vol126 no 5 pp 533ndash538 2000
[18] Y Liu and H-J Li ldquoWave scattering by dual submergedhorizontal porous plates Further resultsrdquo Ocean Engineeringvol 81 no 5 pp 158ndash163 2014
[19] I H Cho and M H Kim ldquoWave absorbing system usinginclined perforated platesrdquo Journal of Fluid Mechanics vol 608pp 1ndash20 2008
[20] K-H Wang and X Ren ldquoWater waves on flexible and porousbreakwatersrdquo Journal of Engineering Mechanics vol 119 no 5pp 1025ndash1047 1993
[21] K H Wang and X Ren ldquoWave interaction with a concentricporous cylinder systemrdquo Ocean Engineering vol 21 no 4 pp343ndash360 1994
[22] F Zhao T Zhang R Wan L Huang X Wang and WBao ldquoHydrodynamic loads acting on a circular porous platehorizontally submerged in wavesrdquo Ocean Engineering vol 136pp 168ndash177 2017
14 Mathematical Problems in Engineering
[23] A Mondal and R Gayen ldquoWave interaction with dual circularporous platesrdquo Journal of Marine Science and Application vol14 no 4 pp 366ndash375 2015
[24] W L Qiao K-H Wang and Y Q Sun ldquoScattering of waterwaves by a floating body with two vertically attached porouswallsrdquo Journal of Engineering Mechanics vol 144 no 2 ArticleID 04017162 2018
[25] S W Twu and D T Lin ldquoOn a highly effective wave absorberrdquoCoastal Engineering Journal vol 15 no 4 pp 389ndash405 1991
[26] X P Yu ldquoDiffraction of water waves by porous breakwatersrdquoJournal of Waterway Port Coastal and Ocean Engineering vol121 no 6 pp 275ndash282 1995
[27] I J Losada M A Losada and A Baquerizo ldquoAn analyticalmethod to evaluate the efficiency of porous screens as wavedampersrdquo Applied Ocean Research vol 15 no 4 pp 207ndash2151993
[28] G Besio and M A Losada ldquoSediment transport patterns atTrafalgar offshore windfarmrdquo Ocean Engineering vol 35 no 7pp 653ndash665 2008
[29] H Song and L Tao ldquoShort-crestedwave interactionwith a con-centric porous cylindrical structurerdquo Applied Ocean Researchvol 29 no 4 pp 199ndash209 2007
[30] L Xiao Y Kou L Tao and L Yang ldquoComparative studyof hydrodynamic performances of breakwaters with double-layered perforated walls attached to ring-shaped very largefloating structuresrdquo Ocean Engineering vol 111 pp 279ndash2912016
[31] T L Yip and A T Chwang ldquoWater wave control by submergedpitching porous platerdquo Journal of Engineering Mechanics vol124 no 4 pp 428ndash434 1998
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6 Mathematical Problems in Engineering
motions across their corresponding porous wall (Chwang[11]) With the application of the linearized Bernoulli equa-tion we have
119882119860 (119911 119905) = minus11986601 [ 1198960120596 (120597120601(119871)1198771120597119905 minus 120597120601(119871)1198772120597119905 )](119871 = 1 2 3) 119909 = minus119886 minus119889 le 119911 le minus119904
(37a)
119882119861 (119911 119905) = minus11986602 [ 1198960120596 (120597120601(119871)1198772120597119905 minus 120597120601(119871)1198773120597119905 )](119871 = 1 2 3) 119909 = 119886 minus119889 le 119911 le minus119904
(37b)
Utilizing the orthogonality property of eigenfunctionscos 119899(119911 + ℎ) for 120601(119871)1198771 and 120601(119871)1198773 (119871 = 1 2 3) over the integralinterval of minusℎ le 119911 le 0 the following equations can beformulated by the matching conditions of (33b) (33c) (34b)(34c) (35b) and (35c) as
int0minusℎ
120597120601(1)1198771120597119909 cos 119899 (119911 + ℎ) 119889119911= intminus119904minusℎ
120597120601(1)1198772120597119909 cos 119899 (119911 + ℎ) 119889119911+ int0minus119904
1 cos 119899 (119911 + ℎ) 119889119911 (119909 = minus119886)(38)
int0minusℎ
120597120601(1)1198773120597119909 cos 119899 (119911 + ℎ) 119889119911= intminus119904minusℎ
120597120601(1)1198772120597119909 cos 119899 (119911 + ℎ) 119889119911+ int0minus119904
1 cos 119899 (119911 + ℎ) 119889119911 (119909 = 119886)(39)
int0minusℎ
120597120601(2)1198771120597119909 cos 119899 (119911 + ℎ) 119889119911= intminus119904minusℎ
120597120601(2)1198772120597119909 cos 119899 (119911 + ℎ) 119889119911 (119909 = minus119886)(40)
int0minusℎ
120597120601(2)1198773120597119909 cos 119899 (119911 + ℎ) 119889119911= intminus119904minusℎ
120597120601(2)1198772120597119909 cos 119899 (119911 + ℎ) 119889119911 (119909 = 119886)(41)
int0minusℎ
120597120601(3)1198771120597119909 cos 119899 (119911 + ℎ) 119889119911= intminus119904minusℎ
120597120601(3)1198772120597119909 cos 119899 (119911 + ℎ) 119889119911+ int0minus119904
(119911 minus 120585119911) cos 119899 (119911 + ℎ) 119889119911 (119909 = minus119886)(42)
int0minusℎ
120597120601(3)1198773120597119909 cos 119899 (119911 + ℎ) 119889119911
= intminus119904minusℎ
120597120601(3)1198772120597119909 cos 119899 (119911 + ℎ) 119889119911+ int0minus119904
(119911 minus 120585119911) cos 119899 (119911 + ℎ) 119889119911 (119909 = 119886)
(43)
Additionally by combining (36a) (36b) (37a) and (37b)with the matching conditions given in (33a) (34a) and(35a) respectively and applying the orthogonality propertyof eigenfunctions cos 119903119899(119911 + ℎ) for 120601(119871)1198772 (L = 1 2 3) over theintegral interval of minusℎ le 119911 le minus119904 we have
intminus119904minusℎ
120601(119871)1198772 cos 119903119899 (119911 + ℎ) 119889119911= intminus119889minusℎ
120601(119871)1198771 cos 119903119899 (119911 + ℎ) 119889119911+ intminus119904minus119889
(119871)1198771 cos 119903119899 (119911 + ℎ) 119889119911(119909 = minus119886 119871 = 1 2 3)
(44)
intminus119904minusℎ
120601(119871)1198772 cos 119903119899 (119911 + ℎ) 119889119911= intminus119889minusℎ
120601(119871)1198773 cos 119903119899 (119911 + ℎ) 119889119911+ intminus119904minus119889
(119871)1198773 cos 119903119899 (119911 + ℎ) 119889119911(119909 = 119886 119871 = 1 2 3)
(45)
where
(1)1198771 = infinsum119898=0
119860(1)119898 (1 + 119898i119866011198960)
cos 119898 (119911 + ℎ)cos 119898ℎ
+ 1i119866011198960
(46)
(2)1198771 = infinsum119898=0
119860(2)119898 (1 + 119898i119866011198960)
cos 119898 (119911 + ℎ)cos 119898ℎ (47)
(3)1198771 = infinsum119898=0
119860(3)119898 (1 + 119898i119866011198960)
cos 119898 (119911 + ℎ)cos 119898ℎ
+ 119911i119866011198960
(48)
Mathematical Problems in Engineering 7
(1)1198773 = infinsum119898=0
119861(1)119898 (1 + 119898i119866021198960)
cos 119898 (119911 + ℎ)cos 119898ℎ
minus 1i119866021198960
(49)
(2)1198773 = infinsum119898=0
119861(2)119898 (1 minus 119898i119866021198960)
cos 119898 (119911 + ℎ)cos 119898ℎ (50)
(3)1198773 = infinsum119898=0
119861(3)119898 (1 + 119898i119866021198960)
cos 119898 (119911 + ℎ)cos 119898ℎ
minus 119911i119866021198960
(51)
Again the summation of terms with the infinite seriesis approximated by a large but finite numbers of termseg a total of M terms Then the unknown coefficients119860(1)119898 119862(1)119898 119864(1)119898 119861(1)119898 (119898 = 0 1 2 119872) [4(M+1) unknownstotally] can be determined by solving the 4(M+1) sys-tem of algebraic equations formulated in (38) (39) (44)and (45) (for L=1) Similarly the unknown coefficients119860(2)119898 119862(2)119898 119864(2)119898 119861(2)119898 (119898 = 0 1 2 119872) can be calculatedfrom the 4(M+1) system of algebraic equations expressedin (40) (41) (44) and (45) (for L=2) while the coefficients119860(3)119898 119862(3)119898 119864(3)119898 119861(3)119898 (119898 = 0 1 2 119872) can be evaluated bysolving the 4(M+1) system of algebraic equations formulatedin (42) (43) (44) and (45) (for L=3) After the determinationof the above unknown coefficients a series of engineeringproperties important to practical applications could be cal-culated for example the hydrodynamic coefficients and themotion induced dynamic responses of the breakwater system
3 Exciting Forces andHydrodynamic Coefficients
31 Exciting Forces To determine the hydrodynamic coeffi-cients related to the inducedmotions of the floating breakwa-ter system the exciting forces and moment of the combinedincident and diffracted waves on the breakwater need tobe computed as inputs for solving the equation of motionBy integrating the dynamic pressures with the use of thevelocity potentials of incident and diffracted waves (see (7)-(9)) over the wet surface the hydrodynamic forces alongthe x direction (1198651198631) and along the z direction (1198651198632) can beformulated respectively as (Qiao et al [24])
1198651198631= i120588120596 infinsum
119898=0
(119868119898 + 119865119898 minus 119879119898) [sin 119898ℎ minus sin 119898 (ℎ minus 119889)]119898 cos 119898ℎ
sdot 119890minusi120596119905 + i120588120596 lowast 2119886 (119889 minus 119904) 1198620119890minusi120596119905minus i120588120596 infinsum
119898=0
2119864119898119903119898 sin 119903119898 (ℎ minus 119889) 119890minusi120596119905(52)
1198651198632 = i120596120588 [21198861198640 + 2 infinsum119898=1
119862119898119903119898 tanh (119903119898119886) cos 119903119898 (ℎ minus 119904)]sdot 119890minusi120596119905
(53)
Also the wave exciting moment (1198651198633) that is referencedto a rotational center (0120585119911) can be derived as
1198651198633 = i120588120596
(119868119898 + 119865119898 minus 119879119898)
sdot [[[[[[
119889 sin 119898 (ℎ minus 119889) minus 120585119911 sin 119898ℎ + 120585119911 sin 119898 (ℎ minus 119889)119898 cos 119898ℎ
+cos 119898ℎ minus cos 119898 (ℎ minus 119889)1198982 cos 119898ℎ
]]]]]]
sdot 119890minusi120596119905 + i120588120596 1198861198620 (1198892 minus 1199042)
minus infinsum119898=1
2119864119898119903119898 [119889 sin 119903119898 (ℎ minus 119889)
+ cos 119903119898 (ℎ minus 119904) minus cos 119903119898 (ℎ minus 119889)119903119898 ] 119890minusi120596119905
+ i120588120596 [21198861198620120585119911 (119889 minus 119904) minus infinsum119898=1
2119864119898120585119911119903119898 sin 119903119898 (ℎ minus 119889)]
sdot 119890minusi120596119905 minus i120588120596 2119862011988633 + infinsum119898=1
[ 2119886119864119898 cos 119903119898 (ℎ minus 119904)119903119898 tanh (119903119898119886)minus 2119864119898 cos 119903119898 (ℎ minus 119904)1199031198982 ] 119890minusi120596119905
(54)
32 Hydrodynamic Coefficients The radiation forces inducedby the translational and rotational motions of the floatingbreakwater system can be calculated from the derived radi-ation potentials utilizing the linearized Bernoulli equation as
119865119871119896 = i120588120596 intΓ
Φ(119871)119877119896 119890minusi120596119905119899119895119889Γ= 12058812059621198830119895 int
Γ120601(119871)119877119896 119890minusi120596119905119899119895119889Γ
= 12059621198830119895119898119871119896119890minusi120596119905 + i1205961198830119895119889119871119896119890minusi120596119905(55)
where 119898119871119896 is the radiation-induced added mass and the119889119871119896 denotes the radiation damping The expressions arerespectively
119898119871119896 = Re(120588 intΓ
120601(119871)119877119896119899119895119889Γ) = Re (120588119871119896) (56)
119889119871119896 = Im (120588120596 intΓ
120601(119871)119877119896119899119895119889Γ) = Im (120588120596119871119896) (57)
8 Mathematical Problems in Engineering
119871119896 can be computed with formulations given below
11 = (int0minus119889
120601(1)1198771 10038161003816100381610038161003816119909=minus119886 119889119911 minus int0minus119889
120601(1)1198773 10038161003816100381610038161003816119909=119886 119889119911)+ (intminus119904minus119889
120601(1)1198772 10038161003816100381610038161003816119909=minus119886 119889119911 minus intminus119904minus119889
120601(1)1198772 10038161003816100381610038161003816119909=119886 119889119911)= 119872sum119898=1
119860(1)119898 minus 119861(1)119898119898 cos 119898ℎ [sin 119898ℎ minus sin 119898 (ℎ minus 119889)]+ 2119886119862(1)0 (119889 minus 119904) + 119872sum
119898=1
2119864(1)119898119903119898 [sin119898120587minus sin 119903119898 (ℎ minus 119889)]
(58)
22 = int119886minus119886
120601(2)1198772 10038161003816100381610038161003816119911=minus119909 119889119909 = 2119886119864(2)0+ 119872sum119898=1
2119862(2)119898119903119898 tanh (119903119898119886) cos 119903119898 (ℎ minus 119904) + 119886 (ℎ minus 119904)minus 11988633 (ℎ minus 119904)
(59)
33 = int0minus119889
( 120601(3)1198771 10038161003816100381610038161003816119909=minus119886 minus 120601(3)1198773 10038161003816100381610038161003816119909=119886) (119911 minus 120585119911) 119889119911+ intminus119904minus119889
(120601(3)1198772 10038161003816100381610038161003816119909=minus119886 minus 120601(3)1198772 10038161003816100381610038161003816119909=119886) (119911 minus 120585119911) 119889119911minus int119886minus119886
120601(3)1198772 10038161003816100381610038161003816119909=minus119904 119909 119889119909 = 119872sum119898=0
(119860(3)119898 minus 119861(3)119898 )
sdot [119889 sin 119898 (ℎ minus 119889) minus 120585119911 sin 119898 (ℎ minus 119889)119898 cos 119898ℎ
+ cos 119898ℎ minus cos 119898 (ℎ minus 119889)1198982 cos 119898ℎ ] + 119886119862(3)0 (1198892 minus 1199042)
minus 119872sum119898=1
2119864(3)119898119903119898 [119889 sin 119898 (ℎ minus 119889)+ cos 119903119898 (ℎ minus 119904) minus cos 119903119898 (ℎ minus 119889)119903119898 ] + 2119886119862(3)0 120585119911 (119889minus 119904) minus 119872sum
119898=1
2119864(3)119898 120585119911119903119898 sin 119903119898 (ℎ minus 119889) minus 23 119862(3)0 1198863 minus 119872sum119898=1
120576sdot cosh 120574119898 (ℎ minus 119904) 1 minus cos 21198861205741198981205741198982 + 119872sum
119898=1
(120576 cos119898120587minus 120576) [ 119889 sinh 120574119898 (ℎ minus 119889) minus 119904 sinh 120574119898 (ℎ minus 119904)120574119898minus cosh 120574119898 (ℎ minus 119904) minus cosh 120574119898 (ℎ minus 119889)1205741198982 ]minus 119872sum119898=1
119864(3)119898 cos 119903119898 (h minus s)119903119898 sinh 119903119898119886 (2119886 cosh 119903119898119886minus 2 sinh 119903119898119886119903119898 )
(60)
31 = 13 = int0minus119889
( 120601(3)1198771 10038161003816100381610038161003816119909=minus119886 minus 120601(3)1198773 10038161003816100381610038161003816119909=119886) 119889119911+ intminus119904minus119889
( 120601(3)1198772 10038161003816100381610038161003816119909=minus119886 minus 120601(3)1198772 10038161003816100381610038161003816119909=119886) 119889119911 = 119872sum119898=0
(119860(3)119898minus 119861(3)119898 ) sin 119898ℎ minus sin 119898 (ℎ minus 119889)
119898 cos 119898ℎ minus 2 (ℎ minus 119904)sdot [ 11988636 (ℎ minus 119904) + 119886119862(3)0 ] + 119886 (ℎ minus 119904)3 minus 119886 (ℎ minus 119889)33 (ℎ minus 119904)+ 119872sum119898=1
2119864(3)119898119903119898 sin 119903119898 (ℎ minus 119889)
(61)
33 Equation of Motion and RAOs The derived solutions forthe diffraction and radiation problems could now be inte-grated with the equation of motion of the floating breakwatersystem to analyze the dynamic responses of the structuralsystem We have
minus1205962 [119872 + 119898] minus i120596 [119889] + [119904] [1198830] = [1198650] (62)
where [1198830] = [11988301 11988302 11988303]119879 and are vectors indicatingthe motion amplitudes and the wave exciting forces respec-tively [119898] and [119889] are addedmass and dampingmatrices [119872]is the mass matrix and [119878] denotes the hydrostatic restoringforce matrix they can be expressed as
[119872] = [[[
1198980 0 minus11989801198851198660 1198980 0minus1198980119885119866 0 1]]]
(63)
[119878] = [[[
0 0 00 2120588119892119886 00 0 1198980119892119866119872
]]]
(64)
Here 1198980 is the mass per unit length of the floatingbreakwater and 119911119866 is the z-coordinate of the center of gravityThe mass moment of inertia about y-axis I can be computedby 119868 = 11989801199031198662 in which 119903119866 is the radius of gyration in pitchingoscillation and 119866119872 is the metacentric height of the floatingbreakwater system Once 1198830119895 (j=1 2 3) are determined bysolving (62) the RAOs (Response Amplitude Operators) ofthe breakwater system can be calculated as
119877119860119874119895 = 1198830119895119867119895 (119895 = 1 2 3) (65)
4 Results and Discussion
The validation of the wave diffraction solutions in terms ofreflectiontransmission coefficients and hydrodynamic loadsfor a train of monochromatic wave propagating throughthe proposed floating breakwater system that consists of apartially submerged rectangular body and with or withoutthe attached two porous side walls has been carried outin a previous study (Qiao et al [24]) Here additional
Mathematical Problems in Engineering 9
Black et al(1971)Present approach
G=G=infinhs=as=
0010203040506070809
F D
gH
a
3 60 5 71 42ks
(a) Horizontal force
Black et al(1971)Present approach
G=G=infinhs= as=
0010203040506070809
1
F D
gHa
3 60 5 71 42ks
(b) Vertical force
Black et al(1971)Present approach
G=G=infinhs= as=
3 60 4 51 72ks
0
005
01
015
02
025
F D
gH
a2
(c) Moment
Figure 2 Comparison of the dimensionless wave exciting forces and moment versus 1198960119904 for ℎ119904 = 20 119886119904 = 30 11986601 = 11986602 997888rarr infin (a)dimensionless horizontal exciting force (b) dimensionless vertical exciting force (c) dimensionless exciting moment about the rotationalcenter of (0 0)
comparisons of wave diffraction results with other publishednumerical solutions for the limiting case of neglecting theeffect of two porous walls are again presented For thewave radiation problem the motion induced hydrodynamiccoefficients such as added mass and damping coefficient arealso investigated for their variations with changes of waveparameters As a validation those results for the caseswithoutporous walls are compared with limited numerical solutionspublished in literatures Subsequently the effects of variousdimensionless parameters for example porous propertiesincident wave conditions and the geometry of floating bodyto wave-induced hydrodynamic forces hydrodynamic coeffi-cients and RAOs were also examined In our computationsthe first 40 terms in the infinite series of the diffractionpotentials and radiation potentials are taken the same as [7]namely M=40
As an example showing the results comparisons for thediffraction problem Figure 2 illustrates the variations ofthe dimensionless exciting forces in horizontal and verticaldirections versus dimensionless wavelength 1198960119904 for the caseof ℎ119904 = 20 119886119904 = 10 11986601 = 11986602 997888rarr infin The excitingmoments with reference to the rotational center of (00) are
also presented in Figure 2 Here 11986601 = 11986602 997888rarr infin representsthe case of a floating breakwater system without dual porousside walls The numerical results computed by Black et al [4]are also plotted in Figure 2 The comparisons indicate thatthe exciting forces and moments are well predicted by thepresent analytical model Both the analytical solutions andthe numerical results reveal that the horizontal exciting forcesand the exciting moments show rapidly increasing trendinitially until a peak value is reached and then are followedby a gradual decrease of the values with the further increaseof 1198960119904 However the vertical forces as shown in Figure 2(b)decrease continuously with an increase in 1198960119904 According tothe results given in the previous study by Qiao et al [24] itindicated that the effectiveness of the porous walls is reflectedwith the reduction of the vertical hydrodynamic forces andwhen the breakwater system is rotated with reference to thebottom center of the body the induced moment can also bereduced However due to the added additional length of theporous walls the horizontal forces are slightly greater thanthose without porous walls
Figure 3 presents the variations of the dimensionlessadded mass and damping coefficients in surge only motion
10 Mathematical Problems in Engineering
YHZheng et al(2004)
G=G=
G=G=
G=G=
G=G=infin
0
1
2
3
4
5
6m
2
as
963 100 87521 114kh
(a) Added mass
YH Zheng et al(2004)
G=G=infin
G=G=
G=G=
G=G=
963 100 87521 114kh
005
115
225
335
4
d
2
as
(b) Radiation coefficient
Figure 3 Variation of the dimensionless surge-surge hydrodynamic coefficients ((a) for the added mass and (b) for the radiation damping)versus 1198960ℎ under the conditions of different porous-effect parameters for 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16
YHZheng et al(2004)
G=G=
G=G=
G=G=
G=G=infin
963 100 87521 114kh
0
02
04
06
08
1
12
14
m
2
as
(a) Added mass
YHZheng et al(2004)
G=G=
G=G=
G=G=
G=G=infin
minus04minus02
002040608
11214
d
2
as
963 100 87521 114kh
(b) Radiation damping
Figure 4 Variation of the dimensionless heave-heave hydrodynamic coefficients ((a) for the added mass and (b) for the radiation damping)versus 1198960ℎ under the conditions of different porous-effect parameters for 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16
versus 1198960ℎ for the setting of 119904ℎ = 13 119889ℎ = 12 and 119886ℎ =16under the conditions of various porous-effect parametersWhen 11986601 = 11986602 = infin it represents the case withoutconsidering the effect of dual porous walls The presentanalytical solutions for this limiting case can be adopted tocompare to the limited numerical results available in thestudy of [7] The comparison plot is shown in Figure 4 wherethe nearly identical matches between the present analyticalsolutions and the numerical results of [7] can be observedThe results shown in Figure 4 suggest that the addition ofporous walls tends to enhance the added mass and dampingcoefficients In general with a decrease of porous effectparameter the addedmass and damping coefficients increaseFor the variation of addedmass coefficient with porouswalls
it can be seen that a peak value occurred at a long wave (small1198960ℎ value) condition However for the radiation dampingcoefficient it is shown to increase rapidly to its maximumvalue with an increase of 1198960ℎ then a gradual decrease trendis followed by a further increase of 1198960ℎ
The dimensionless heave-heave added mass (119898222120588119886119904)and radiation damping coefficient (119889222120588120596119886119904) versus 1198960ℎunder various 11986601 and 11986602 values for the case of 119904ℎ =13 119889ℎ = 12 119886ℎ = 16 are presented in Figure 4Under the limiting condition of 11986601 = 11986602 = infin (withoutporous walls) the numerical results produced by the study of[7] are also plotted for comparison Again good agreementsbetween the present analytical solutions and the publishednumerical results can be noticed The results in Figure 4(a)
Mathematical Problems in Engineering 11
YHZheng et al(2004)
963 100 87521 114kh
0
02
04
06
08
1
12
14m
2
as
G=G=infin
G=G=
G=G=
G=G=
(a) Added mass
YH Zhang et al(2004)
G=G=infin
G=G=
G=G=
G=G=
0010203040506070809
d
2
as
963 100 87521 114kh
(b) Radiation damping
Figure 5 Variation of the dimensionless pitch-pitch hydrodynamic coefficients ((a) for the added mass and (b) for the radiation damping)versus 1198960ℎ under the conditions of different porous-effect parameters for 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16 with the rotational center set at (00)
suggest that the variations of added mass for the shorterwavelength waves (large 1198960ℎ values) are less sensitive to thechange of porous parameters The presence of dual porouswalls increases the heave-heave added mass which tends toreach its maximum value at the long wave extreme Alsoa decrease in porous parameters results in an increase inheave induced added mass As shown in Figure 4(b) theheave-heave damping coefficients decrease rapidly when 1198960ℎincreases initially Then a gradual changing trend is followedby a further increase in 1198960ℎ Additionally the presence ofdual porous walls tends to reduce the values of heave-heavedamping coefficients
For the pitching induced added mass and dampingcoefficients the results are plotted against 1198960ℎ in Figure 5for various porous-effect parameters 11986601 and 11986602 and underthe settings of 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16 Whenthe rotational center is set at (00) as shown in Figure 5 thepresent analytical solutions match well with the publishednumerical results [7] for the limiting case of no porouswall (ie 11986601 = 11986602 = infin) It can be observed thatthe pitching oscillation induced added mass and dampingcoefficients for the floating breakwater system with dualporous side walls are greater than those without porous wallsWith decrease of the porous effect parameters the addedmass and damping coefficients increase For the added masscoefficients it is found that the variations trends are morecomplex for waves with relatively smaller 1198960ℎ under theconditions of different porous-effect parameters Generallyin cases of weaker porous-effect (for example 11986601 = 11986602 le 2as shown in Figure 5(a)) the value of added mass coefficienttends to approach 14 under the case considered For thevariations of damping coefficient as shown in Figure 5(b) thevarying trends are similar to those of damping coefficient insurge motion
The results presented in Figure 6 are the variations ofthe dimensionless surge-pitch added mass and damping
coefficients versus 1198960ℎ for varying 11986601 and 11986602 The case isfor 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16 with referenceto the rotational center of (00) Clearly the added massand damping coefficients are shown to increase with anincrease in the porous-effect parameters of both porouswalls A and B while the damping coefficients reach theirabsolute maximum values when 1198960ℎ is within the rangeof 1 and 2 The magnitude of surge-pitch hydrodynamiccoefficients especially the damping is greatly influenced bythe permeability of the dual porous walls compared to thesituation with surge or pitching motion only
The analytical solutions for the Response AmplitudeOperators (RAOs) of the floating breakwater system pro-posed in this study are illustrated in Figure 7 as a functionof dimensionless wavelength 1198960ℎ for various porous-effectparameters with the conditions of 119904ℎ = 03 119889ℎ = 05119886ℎ = 025 119903119866119886 = 09 and 119866119872119886 = 01 where therotational center is located at (00) It is found that the varyingtrends of surge RAOs are similar to those of pitching RAOsalthough the peak values of surge RAOs are smaller thanthose of pitching RAOs When the porous-effect parametersdecrease the main peak values of either surge or pitchingRAOs that occurred are shifted to the wave conditions withshorter wavelength Furthermore for the heave RAO itexperiences a typical damped response for the case of singledegree of freedom system and its maximum values of RAOwould move to the lower frequency wave region when theporous-effect parameters decrease
5 Summary and Conclusions
The hydrodynamic performance investigations includingthe hydrodynamic loads radiation induced added massand damping coefficients and dynamic responses relatedRAOs are critical to the study that is associated with theproblems of wave and structure interactions In this study the
12 Mathematical Problems in Engineering
G=G=infin
G=G=
G=G=
G=G=
963 100 87521 114kh
minus4
minus3
minus2
minus1
0
1
m
2
as
(a) Added mass
G=G=infin
G=G=
G=G=
G=G=
963 100 87521 114kh
minus3
minus25
minus2
minus15
minus1
minus05
0
d
2
as
(b) Radiation damping
Figure 6 Variation of the dimensionless surge-pitch hydrodynamic coefficients ((a) for the added mass and (b) for the radiation damping)versus 1198960ℎ under the conditions of different porous-effect parameters for 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16 with the rotational center set at(0 0)
G=G=
G=G=
G=G=
G=G=infin
4515 3525 3 605 2 5 551 4kh
0123456789
1011
|X|A
(a) Surge RAO
G=G=infin
G=G=
G=G=
G=G=
3 60 4 51 2kh
0
05
1
15
2
25
3
35|X
|A
(b) Heave RAO
G=G=
G=G=
G=G=
G=G=infin
05 3515 325 45 65 5541 20kh
0
5
10
15
|X|A
(c) Pitch RAO
Figure 7 Variation of the dimensionless RAO ((a) for surge RAO (b) for heave RAO and (c) for pitch RAO) versus 1198960ℎ under the conditionsof different porous-effect parameters for 119904ℎ = 03 119889ℎ = 05 119886ℎ = 025 119903119866119886 = 09 119866119872119886 = 01 with rotation center at (00)
Mathematical Problems in Engineering 13
hydrodynamic problems considering a floating breakwatersystem that consists of a rectangular body and two verticalattached porous walls are investigated analytically based onthe two-dimensional potential flow theory The expressionsof the diffraction and radiation based velocity potentials andthe associated unknown coefficients for each subdomain arederived by the MEEM approach Subsequently the excitinghydrodynamic forces added mass and radiation dampingcoefficients in terms of surge heave and pitching motionsand the equation ofmotion for the floating breakwater systemare formulated and calculated for various case scenariosby varying the wave conditions and porous effect param-eters Considering the limiting case without porous wallsgood agreements between the present analytical solutionsand published numerical results are achieved This suggeststhat the present analytical model is demonstrated to be areliable engineering tool to provide reasonable estimationsfor the hydrodynamic performances of a floating breakwatersystem with dual side porous walls It is worthy to noticethat the wave loads on the floating structures especiallythe moments when referenced to the rotational center ofbottom center would be reduced with an increase in theporous-effect parameters of the porous walls Additionallythe existence of the dual porous walls could move the peaksof surge and pitching RAOs to the waves of higher frequencywhen the porous effect parameters decrease Overall it canbe concluded that a properly adopted porous-wall systemwhen attached to a floating body can effectively improveits hydrodynamic performance as a breakwater The analyt-ical approach developed in this study can also be furtherextended to assist the theoretical analysis for other similar butimproved floating breakwater systems
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was supported by ldquoFundamental Research Fundsfor the Central Universitiesrdquo (Grand No 017180224)
References
[1] I-H Cho ldquoTransmission coefficients of a floating rectangularbreakwater with porous side platesrdquo International Journal ofNaval Architecture and Ocean Engineering vol 8 no 1 pp 53ndash65 2016
[2] H Hu K-H Wang and A N Williams ldquoWave motion over abreakwater system of a horizontal plate and a vertical porouswallrdquo Ocean Engineering vol 29 no 4 pp 373ndash386 2001
[3] C C Mei and J L Black ldquoScattering of surface waves byrectangular obstacles in water of finite depthrdquo Journal of FluidMechanics vol 38 no 3 pp 499ndash511 1969
[4] J L Black C C Mei and M C G Bray ldquoRadiation andscattering of water waves by rigid bodiesrdquo Journal of FluidMechanics vol 46 no 1 pp 151ndash164 1971
[5] N Drimer Y Agnon and M Stiassnie ldquoA simplified analyticalmodel for a floating breakwater in water of finite depthrdquoAppliedOcean Research vol 14 no 1 pp 33ndash41 1992
[6] J-F Lee ldquoOn the heave radiation of a rectangular structurerdquoOcean Engineering vol 22 no 1 pp 19ndash34 1995
[7] Y H Zheng Y G You and Y M Shen ldquoOn the radiationand diffraction of water waves by a rectangular buoyrdquo OceanEngineering vol 31 no 8-9 pp 1063ndash1082 2004
[8] E D Christensen H B Bingham A P Skou Friis A K Larsenand K L Jensen ldquoAn experimental and numerical study offloating breakwatersrdquo Coastal Engineering Journal vol 137 pp43ndash58 2018
[9] E Koutandos P Prinos andXGironella ldquoFloating breakwatersunder regular and irregular wave forcing Reflection and trans-mission characteristicsrdquo Journal of Hydraulic Research vol 43no 2 pp 174ndash188 2005
[10] G H Dong Y N Zheng Y C Li B Teng C T Guan and D FLin ldquoExperiments on wave transmission coefficients of floatingbreakwatersrdquo Ocean Engineering vol 35 no 8-9 pp 931ndash9382008
[11] A T Chwang ldquoA porous-wavemaker theoryrdquo Journal of FluidMechanics vol 132 pp 395ndash406 1983
[12] Y C Li Y Liu and B Teng ldquoPorous effect parameter of thinpermeable platesrdquo Coastal Engineering Journal vol 48 no 4pp 309ndash336 2006
[13] S R Manam and M Sivanesan ldquoScattering of water wavesby vertical porous barriers an analytical approachrdquo WaveMotion An International Journal Reporting Research on WavePhenomena vol 67 pp 89ndash101 2016
[14] A T Chwang and Z Dong ldquoWave-trapping due to a porousplaterdquo in Proceedings of 15 ONR symposium on Naval Hydrody-namics pp 407ndash414 1985
[15] S Zhu and A T Chwang ldquoAnalytical study of porous waveabsorberrdquo Journal of Engineering Mechanics vol 127 no 4 pp326ndash332 2001
[16] H Behera and T Sahoo ldquoHydroelastic analysis of gravity waveinteraction with submerged horizontal flexible porous platerdquoJournal of Fluids and Structures vol 54 pp 643ndash660 2015
[17] T L Yip and A T Chwang ldquoPerforated wall breakwater withinternal horizontal platerdquo Journal of Engineering Mechanics vol126 no 5 pp 533ndash538 2000
[18] Y Liu and H-J Li ldquoWave scattering by dual submergedhorizontal porous plates Further resultsrdquo Ocean Engineeringvol 81 no 5 pp 158ndash163 2014
[19] I H Cho and M H Kim ldquoWave absorbing system usinginclined perforated platesrdquo Journal of Fluid Mechanics vol 608pp 1ndash20 2008
[20] K-H Wang and X Ren ldquoWater waves on flexible and porousbreakwatersrdquo Journal of Engineering Mechanics vol 119 no 5pp 1025ndash1047 1993
[21] K H Wang and X Ren ldquoWave interaction with a concentricporous cylinder systemrdquo Ocean Engineering vol 21 no 4 pp343ndash360 1994
[22] F Zhao T Zhang R Wan L Huang X Wang and WBao ldquoHydrodynamic loads acting on a circular porous platehorizontally submerged in wavesrdquo Ocean Engineering vol 136pp 168ndash177 2017
14 Mathematical Problems in Engineering
[23] A Mondal and R Gayen ldquoWave interaction with dual circularporous platesrdquo Journal of Marine Science and Application vol14 no 4 pp 366ndash375 2015
[24] W L Qiao K-H Wang and Y Q Sun ldquoScattering of waterwaves by a floating body with two vertically attached porouswallsrdquo Journal of Engineering Mechanics vol 144 no 2 ArticleID 04017162 2018
[25] S W Twu and D T Lin ldquoOn a highly effective wave absorberrdquoCoastal Engineering Journal vol 15 no 4 pp 389ndash405 1991
[26] X P Yu ldquoDiffraction of water waves by porous breakwatersrdquoJournal of Waterway Port Coastal and Ocean Engineering vol121 no 6 pp 275ndash282 1995
[27] I J Losada M A Losada and A Baquerizo ldquoAn analyticalmethod to evaluate the efficiency of porous screens as wavedampersrdquo Applied Ocean Research vol 15 no 4 pp 207ndash2151993
[28] G Besio and M A Losada ldquoSediment transport patterns atTrafalgar offshore windfarmrdquo Ocean Engineering vol 35 no 7pp 653ndash665 2008
[29] H Song and L Tao ldquoShort-crestedwave interactionwith a con-centric porous cylindrical structurerdquo Applied Ocean Researchvol 29 no 4 pp 199ndash209 2007
[30] L Xiao Y Kou L Tao and L Yang ldquoComparative studyof hydrodynamic performances of breakwaters with double-layered perforated walls attached to ring-shaped very largefloating structuresrdquo Ocean Engineering vol 111 pp 279ndash2912016
[31] T L Yip and A T Chwang ldquoWater wave control by submergedpitching porous platerdquo Journal of Engineering Mechanics vol124 no 4 pp 428ndash434 1998
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Mathematical Problems in Engineering 7
(1)1198773 = infinsum119898=0
119861(1)119898 (1 + 119898i119866021198960)
cos 119898 (119911 + ℎ)cos 119898ℎ
minus 1i119866021198960
(49)
(2)1198773 = infinsum119898=0
119861(2)119898 (1 minus 119898i119866021198960)
cos 119898 (119911 + ℎ)cos 119898ℎ (50)
(3)1198773 = infinsum119898=0
119861(3)119898 (1 + 119898i119866021198960)
cos 119898 (119911 + ℎ)cos 119898ℎ
minus 119911i119866021198960
(51)
Again the summation of terms with the infinite seriesis approximated by a large but finite numbers of termseg a total of M terms Then the unknown coefficients119860(1)119898 119862(1)119898 119864(1)119898 119861(1)119898 (119898 = 0 1 2 119872) [4(M+1) unknownstotally] can be determined by solving the 4(M+1) sys-tem of algebraic equations formulated in (38) (39) (44)and (45) (for L=1) Similarly the unknown coefficients119860(2)119898 119862(2)119898 119864(2)119898 119861(2)119898 (119898 = 0 1 2 119872) can be calculatedfrom the 4(M+1) system of algebraic equations expressedin (40) (41) (44) and (45) (for L=2) while the coefficients119860(3)119898 119862(3)119898 119864(3)119898 119861(3)119898 (119898 = 0 1 2 119872) can be evaluated bysolving the 4(M+1) system of algebraic equations formulatedin (42) (43) (44) and (45) (for L=3) After the determinationof the above unknown coefficients a series of engineeringproperties important to practical applications could be cal-culated for example the hydrodynamic coefficients and themotion induced dynamic responses of the breakwater system
3 Exciting Forces andHydrodynamic Coefficients
31 Exciting Forces To determine the hydrodynamic coeffi-cients related to the inducedmotions of the floating breakwa-ter system the exciting forces and moment of the combinedincident and diffracted waves on the breakwater need tobe computed as inputs for solving the equation of motionBy integrating the dynamic pressures with the use of thevelocity potentials of incident and diffracted waves (see (7)-(9)) over the wet surface the hydrodynamic forces alongthe x direction (1198651198631) and along the z direction (1198651198632) can beformulated respectively as (Qiao et al [24])
1198651198631= i120588120596 infinsum
119898=0
(119868119898 + 119865119898 minus 119879119898) [sin 119898ℎ minus sin 119898 (ℎ minus 119889)]119898 cos 119898ℎ
sdot 119890minusi120596119905 + i120588120596 lowast 2119886 (119889 minus 119904) 1198620119890minusi120596119905minus i120588120596 infinsum
119898=0
2119864119898119903119898 sin 119903119898 (ℎ minus 119889) 119890minusi120596119905(52)
1198651198632 = i120596120588 [21198861198640 + 2 infinsum119898=1
119862119898119903119898 tanh (119903119898119886) cos 119903119898 (ℎ minus 119904)]sdot 119890minusi120596119905
(53)
Also the wave exciting moment (1198651198633) that is referencedto a rotational center (0120585119911) can be derived as
1198651198633 = i120588120596
(119868119898 + 119865119898 minus 119879119898)
sdot [[[[[[
119889 sin 119898 (ℎ minus 119889) minus 120585119911 sin 119898ℎ + 120585119911 sin 119898 (ℎ minus 119889)119898 cos 119898ℎ
+cos 119898ℎ minus cos 119898 (ℎ minus 119889)1198982 cos 119898ℎ
]]]]]]
sdot 119890minusi120596119905 + i120588120596 1198861198620 (1198892 minus 1199042)
minus infinsum119898=1
2119864119898119903119898 [119889 sin 119903119898 (ℎ minus 119889)
+ cos 119903119898 (ℎ minus 119904) minus cos 119903119898 (ℎ minus 119889)119903119898 ] 119890minusi120596119905
+ i120588120596 [21198861198620120585119911 (119889 minus 119904) minus infinsum119898=1
2119864119898120585119911119903119898 sin 119903119898 (ℎ minus 119889)]
sdot 119890minusi120596119905 minus i120588120596 2119862011988633 + infinsum119898=1
[ 2119886119864119898 cos 119903119898 (ℎ minus 119904)119903119898 tanh (119903119898119886)minus 2119864119898 cos 119903119898 (ℎ minus 119904)1199031198982 ] 119890minusi120596119905
(54)
32 Hydrodynamic Coefficients The radiation forces inducedby the translational and rotational motions of the floatingbreakwater system can be calculated from the derived radi-ation potentials utilizing the linearized Bernoulli equation as
119865119871119896 = i120588120596 intΓ
Φ(119871)119877119896 119890minusi120596119905119899119895119889Γ= 12058812059621198830119895 int
Γ120601(119871)119877119896 119890minusi120596119905119899119895119889Γ
= 12059621198830119895119898119871119896119890minusi120596119905 + i1205961198830119895119889119871119896119890minusi120596119905(55)
where 119898119871119896 is the radiation-induced added mass and the119889119871119896 denotes the radiation damping The expressions arerespectively
119898119871119896 = Re(120588 intΓ
120601(119871)119877119896119899119895119889Γ) = Re (120588119871119896) (56)
119889119871119896 = Im (120588120596 intΓ
120601(119871)119877119896119899119895119889Γ) = Im (120588120596119871119896) (57)
8 Mathematical Problems in Engineering
119871119896 can be computed with formulations given below
11 = (int0minus119889
120601(1)1198771 10038161003816100381610038161003816119909=minus119886 119889119911 minus int0minus119889
120601(1)1198773 10038161003816100381610038161003816119909=119886 119889119911)+ (intminus119904minus119889
120601(1)1198772 10038161003816100381610038161003816119909=minus119886 119889119911 minus intminus119904minus119889
120601(1)1198772 10038161003816100381610038161003816119909=119886 119889119911)= 119872sum119898=1
119860(1)119898 minus 119861(1)119898119898 cos 119898ℎ [sin 119898ℎ minus sin 119898 (ℎ minus 119889)]+ 2119886119862(1)0 (119889 minus 119904) + 119872sum
119898=1
2119864(1)119898119903119898 [sin119898120587minus sin 119903119898 (ℎ minus 119889)]
(58)
22 = int119886minus119886
120601(2)1198772 10038161003816100381610038161003816119911=minus119909 119889119909 = 2119886119864(2)0+ 119872sum119898=1
2119862(2)119898119903119898 tanh (119903119898119886) cos 119903119898 (ℎ minus 119904) + 119886 (ℎ minus 119904)minus 11988633 (ℎ minus 119904)
(59)
33 = int0minus119889
( 120601(3)1198771 10038161003816100381610038161003816119909=minus119886 minus 120601(3)1198773 10038161003816100381610038161003816119909=119886) (119911 minus 120585119911) 119889119911+ intminus119904minus119889
(120601(3)1198772 10038161003816100381610038161003816119909=minus119886 minus 120601(3)1198772 10038161003816100381610038161003816119909=119886) (119911 minus 120585119911) 119889119911minus int119886minus119886
120601(3)1198772 10038161003816100381610038161003816119909=minus119904 119909 119889119909 = 119872sum119898=0
(119860(3)119898 minus 119861(3)119898 )
sdot [119889 sin 119898 (ℎ minus 119889) minus 120585119911 sin 119898 (ℎ minus 119889)119898 cos 119898ℎ
+ cos 119898ℎ minus cos 119898 (ℎ minus 119889)1198982 cos 119898ℎ ] + 119886119862(3)0 (1198892 minus 1199042)
minus 119872sum119898=1
2119864(3)119898119903119898 [119889 sin 119898 (ℎ minus 119889)+ cos 119903119898 (ℎ minus 119904) minus cos 119903119898 (ℎ minus 119889)119903119898 ] + 2119886119862(3)0 120585119911 (119889minus 119904) minus 119872sum
119898=1
2119864(3)119898 120585119911119903119898 sin 119903119898 (ℎ minus 119889) minus 23 119862(3)0 1198863 minus 119872sum119898=1
120576sdot cosh 120574119898 (ℎ minus 119904) 1 minus cos 21198861205741198981205741198982 + 119872sum
119898=1
(120576 cos119898120587minus 120576) [ 119889 sinh 120574119898 (ℎ minus 119889) minus 119904 sinh 120574119898 (ℎ minus 119904)120574119898minus cosh 120574119898 (ℎ minus 119904) minus cosh 120574119898 (ℎ minus 119889)1205741198982 ]minus 119872sum119898=1
119864(3)119898 cos 119903119898 (h minus s)119903119898 sinh 119903119898119886 (2119886 cosh 119903119898119886minus 2 sinh 119903119898119886119903119898 )
(60)
31 = 13 = int0minus119889
( 120601(3)1198771 10038161003816100381610038161003816119909=minus119886 minus 120601(3)1198773 10038161003816100381610038161003816119909=119886) 119889119911+ intminus119904minus119889
( 120601(3)1198772 10038161003816100381610038161003816119909=minus119886 minus 120601(3)1198772 10038161003816100381610038161003816119909=119886) 119889119911 = 119872sum119898=0
(119860(3)119898minus 119861(3)119898 ) sin 119898ℎ minus sin 119898 (ℎ minus 119889)
119898 cos 119898ℎ minus 2 (ℎ minus 119904)sdot [ 11988636 (ℎ minus 119904) + 119886119862(3)0 ] + 119886 (ℎ minus 119904)3 minus 119886 (ℎ minus 119889)33 (ℎ minus 119904)+ 119872sum119898=1
2119864(3)119898119903119898 sin 119903119898 (ℎ minus 119889)
(61)
33 Equation of Motion and RAOs The derived solutions forthe diffraction and radiation problems could now be inte-grated with the equation of motion of the floating breakwatersystem to analyze the dynamic responses of the structuralsystem We have
minus1205962 [119872 + 119898] minus i120596 [119889] + [119904] [1198830] = [1198650] (62)
where [1198830] = [11988301 11988302 11988303]119879 and are vectors indicatingthe motion amplitudes and the wave exciting forces respec-tively [119898] and [119889] are addedmass and dampingmatrices [119872]is the mass matrix and [119878] denotes the hydrostatic restoringforce matrix they can be expressed as
[119872] = [[[
1198980 0 minus11989801198851198660 1198980 0minus1198980119885119866 0 1]]]
(63)
[119878] = [[[
0 0 00 2120588119892119886 00 0 1198980119892119866119872
]]]
(64)
Here 1198980 is the mass per unit length of the floatingbreakwater and 119911119866 is the z-coordinate of the center of gravityThe mass moment of inertia about y-axis I can be computedby 119868 = 11989801199031198662 in which 119903119866 is the radius of gyration in pitchingoscillation and 119866119872 is the metacentric height of the floatingbreakwater system Once 1198830119895 (j=1 2 3) are determined bysolving (62) the RAOs (Response Amplitude Operators) ofthe breakwater system can be calculated as
119877119860119874119895 = 1198830119895119867119895 (119895 = 1 2 3) (65)
4 Results and Discussion
The validation of the wave diffraction solutions in terms ofreflectiontransmission coefficients and hydrodynamic loadsfor a train of monochromatic wave propagating throughthe proposed floating breakwater system that consists of apartially submerged rectangular body and with or withoutthe attached two porous side walls has been carried outin a previous study (Qiao et al [24]) Here additional
Mathematical Problems in Engineering 9
Black et al(1971)Present approach
G=G=infinhs=as=
0010203040506070809
F D
gH
a
3 60 5 71 42ks
(a) Horizontal force
Black et al(1971)Present approach
G=G=infinhs= as=
0010203040506070809
1
F D
gHa
3 60 5 71 42ks
(b) Vertical force
Black et al(1971)Present approach
G=G=infinhs= as=
3 60 4 51 72ks
0
005
01
015
02
025
F D
gH
a2
(c) Moment
Figure 2 Comparison of the dimensionless wave exciting forces and moment versus 1198960119904 for ℎ119904 = 20 119886119904 = 30 11986601 = 11986602 997888rarr infin (a)dimensionless horizontal exciting force (b) dimensionless vertical exciting force (c) dimensionless exciting moment about the rotationalcenter of (0 0)
comparisons of wave diffraction results with other publishednumerical solutions for the limiting case of neglecting theeffect of two porous walls are again presented For thewave radiation problem the motion induced hydrodynamiccoefficients such as added mass and damping coefficient arealso investigated for their variations with changes of waveparameters As a validation those results for the caseswithoutporous walls are compared with limited numerical solutionspublished in literatures Subsequently the effects of variousdimensionless parameters for example porous propertiesincident wave conditions and the geometry of floating bodyto wave-induced hydrodynamic forces hydrodynamic coeffi-cients and RAOs were also examined In our computationsthe first 40 terms in the infinite series of the diffractionpotentials and radiation potentials are taken the same as [7]namely M=40
As an example showing the results comparisons for thediffraction problem Figure 2 illustrates the variations ofthe dimensionless exciting forces in horizontal and verticaldirections versus dimensionless wavelength 1198960119904 for the caseof ℎ119904 = 20 119886119904 = 10 11986601 = 11986602 997888rarr infin The excitingmoments with reference to the rotational center of (00) are
also presented in Figure 2 Here 11986601 = 11986602 997888rarr infin representsthe case of a floating breakwater system without dual porousside walls The numerical results computed by Black et al [4]are also plotted in Figure 2 The comparisons indicate thatthe exciting forces and moments are well predicted by thepresent analytical model Both the analytical solutions andthe numerical results reveal that the horizontal exciting forcesand the exciting moments show rapidly increasing trendinitially until a peak value is reached and then are followedby a gradual decrease of the values with the further increaseof 1198960119904 However the vertical forces as shown in Figure 2(b)decrease continuously with an increase in 1198960119904 According tothe results given in the previous study by Qiao et al [24] itindicated that the effectiveness of the porous walls is reflectedwith the reduction of the vertical hydrodynamic forces andwhen the breakwater system is rotated with reference to thebottom center of the body the induced moment can also bereduced However due to the added additional length of theporous walls the horizontal forces are slightly greater thanthose without porous walls
Figure 3 presents the variations of the dimensionlessadded mass and damping coefficients in surge only motion
10 Mathematical Problems in Engineering
YHZheng et al(2004)
G=G=
G=G=
G=G=
G=G=infin
0
1
2
3
4
5
6m
2
as
963 100 87521 114kh
(a) Added mass
YH Zheng et al(2004)
G=G=infin
G=G=
G=G=
G=G=
963 100 87521 114kh
005
115
225
335
4
d
2
as
(b) Radiation coefficient
Figure 3 Variation of the dimensionless surge-surge hydrodynamic coefficients ((a) for the added mass and (b) for the radiation damping)versus 1198960ℎ under the conditions of different porous-effect parameters for 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16
YHZheng et al(2004)
G=G=
G=G=
G=G=
G=G=infin
963 100 87521 114kh
0
02
04
06
08
1
12
14
m
2
as
(a) Added mass
YHZheng et al(2004)
G=G=
G=G=
G=G=
G=G=infin
minus04minus02
002040608
11214
d
2
as
963 100 87521 114kh
(b) Radiation damping
Figure 4 Variation of the dimensionless heave-heave hydrodynamic coefficients ((a) for the added mass and (b) for the radiation damping)versus 1198960ℎ under the conditions of different porous-effect parameters for 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16
versus 1198960ℎ for the setting of 119904ℎ = 13 119889ℎ = 12 and 119886ℎ =16under the conditions of various porous-effect parametersWhen 11986601 = 11986602 = infin it represents the case withoutconsidering the effect of dual porous walls The presentanalytical solutions for this limiting case can be adopted tocompare to the limited numerical results available in thestudy of [7] The comparison plot is shown in Figure 4 wherethe nearly identical matches between the present analyticalsolutions and the numerical results of [7] can be observedThe results shown in Figure 4 suggest that the addition ofporous walls tends to enhance the added mass and dampingcoefficients In general with a decrease of porous effectparameter the addedmass and damping coefficients increaseFor the variation of addedmass coefficient with porouswalls
it can be seen that a peak value occurred at a long wave (small1198960ℎ value) condition However for the radiation dampingcoefficient it is shown to increase rapidly to its maximumvalue with an increase of 1198960ℎ then a gradual decrease trendis followed by a further increase of 1198960ℎ
The dimensionless heave-heave added mass (119898222120588119886119904)and radiation damping coefficient (119889222120588120596119886119904) versus 1198960ℎunder various 11986601 and 11986602 values for the case of 119904ℎ =13 119889ℎ = 12 119886ℎ = 16 are presented in Figure 4Under the limiting condition of 11986601 = 11986602 = infin (withoutporous walls) the numerical results produced by the study of[7] are also plotted for comparison Again good agreementsbetween the present analytical solutions and the publishednumerical results can be noticed The results in Figure 4(a)
Mathematical Problems in Engineering 11
YHZheng et al(2004)
963 100 87521 114kh
0
02
04
06
08
1
12
14m
2
as
G=G=infin
G=G=
G=G=
G=G=
(a) Added mass
YH Zhang et al(2004)
G=G=infin
G=G=
G=G=
G=G=
0010203040506070809
d
2
as
963 100 87521 114kh
(b) Radiation damping
Figure 5 Variation of the dimensionless pitch-pitch hydrodynamic coefficients ((a) for the added mass and (b) for the radiation damping)versus 1198960ℎ under the conditions of different porous-effect parameters for 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16 with the rotational center set at (00)
suggest that the variations of added mass for the shorterwavelength waves (large 1198960ℎ values) are less sensitive to thechange of porous parameters The presence of dual porouswalls increases the heave-heave added mass which tends toreach its maximum value at the long wave extreme Alsoa decrease in porous parameters results in an increase inheave induced added mass As shown in Figure 4(b) theheave-heave damping coefficients decrease rapidly when 1198960ℎincreases initially Then a gradual changing trend is followedby a further increase in 1198960ℎ Additionally the presence ofdual porous walls tends to reduce the values of heave-heavedamping coefficients
For the pitching induced added mass and dampingcoefficients the results are plotted against 1198960ℎ in Figure 5for various porous-effect parameters 11986601 and 11986602 and underthe settings of 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16 Whenthe rotational center is set at (00) as shown in Figure 5 thepresent analytical solutions match well with the publishednumerical results [7] for the limiting case of no porouswall (ie 11986601 = 11986602 = infin) It can be observed thatthe pitching oscillation induced added mass and dampingcoefficients for the floating breakwater system with dualporous side walls are greater than those without porous wallsWith decrease of the porous effect parameters the addedmass and damping coefficients increase For the added masscoefficients it is found that the variations trends are morecomplex for waves with relatively smaller 1198960ℎ under theconditions of different porous-effect parameters Generallyin cases of weaker porous-effect (for example 11986601 = 11986602 le 2as shown in Figure 5(a)) the value of added mass coefficienttends to approach 14 under the case considered For thevariations of damping coefficient as shown in Figure 5(b) thevarying trends are similar to those of damping coefficient insurge motion
The results presented in Figure 6 are the variations ofthe dimensionless surge-pitch added mass and damping
coefficients versus 1198960ℎ for varying 11986601 and 11986602 The case isfor 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16 with referenceto the rotational center of (00) Clearly the added massand damping coefficients are shown to increase with anincrease in the porous-effect parameters of both porouswalls A and B while the damping coefficients reach theirabsolute maximum values when 1198960ℎ is within the rangeof 1 and 2 The magnitude of surge-pitch hydrodynamiccoefficients especially the damping is greatly influenced bythe permeability of the dual porous walls compared to thesituation with surge or pitching motion only
The analytical solutions for the Response AmplitudeOperators (RAOs) of the floating breakwater system pro-posed in this study are illustrated in Figure 7 as a functionof dimensionless wavelength 1198960ℎ for various porous-effectparameters with the conditions of 119904ℎ = 03 119889ℎ = 05119886ℎ = 025 119903119866119886 = 09 and 119866119872119886 = 01 where therotational center is located at (00) It is found that the varyingtrends of surge RAOs are similar to those of pitching RAOsalthough the peak values of surge RAOs are smaller thanthose of pitching RAOs When the porous-effect parametersdecrease the main peak values of either surge or pitchingRAOs that occurred are shifted to the wave conditions withshorter wavelength Furthermore for the heave RAO itexperiences a typical damped response for the case of singledegree of freedom system and its maximum values of RAOwould move to the lower frequency wave region when theporous-effect parameters decrease
5 Summary and Conclusions
The hydrodynamic performance investigations includingthe hydrodynamic loads radiation induced added massand damping coefficients and dynamic responses relatedRAOs are critical to the study that is associated with theproblems of wave and structure interactions In this study the
12 Mathematical Problems in Engineering
G=G=infin
G=G=
G=G=
G=G=
963 100 87521 114kh
minus4
minus3
minus2
minus1
0
1
m
2
as
(a) Added mass
G=G=infin
G=G=
G=G=
G=G=
963 100 87521 114kh
minus3
minus25
minus2
minus15
minus1
minus05
0
d
2
as
(b) Radiation damping
Figure 6 Variation of the dimensionless surge-pitch hydrodynamic coefficients ((a) for the added mass and (b) for the radiation damping)versus 1198960ℎ under the conditions of different porous-effect parameters for 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16 with the rotational center set at(0 0)
G=G=
G=G=
G=G=
G=G=infin
4515 3525 3 605 2 5 551 4kh
0123456789
1011
|X|A
(a) Surge RAO
G=G=infin
G=G=
G=G=
G=G=
3 60 4 51 2kh
0
05
1
15
2
25
3
35|X
|A
(b) Heave RAO
G=G=
G=G=
G=G=
G=G=infin
05 3515 325 45 65 5541 20kh
0
5
10
15
|X|A
(c) Pitch RAO
Figure 7 Variation of the dimensionless RAO ((a) for surge RAO (b) for heave RAO and (c) for pitch RAO) versus 1198960ℎ under the conditionsof different porous-effect parameters for 119904ℎ = 03 119889ℎ = 05 119886ℎ = 025 119903119866119886 = 09 119866119872119886 = 01 with rotation center at (00)
Mathematical Problems in Engineering 13
hydrodynamic problems considering a floating breakwatersystem that consists of a rectangular body and two verticalattached porous walls are investigated analytically based onthe two-dimensional potential flow theory The expressionsof the diffraction and radiation based velocity potentials andthe associated unknown coefficients for each subdomain arederived by the MEEM approach Subsequently the excitinghydrodynamic forces added mass and radiation dampingcoefficients in terms of surge heave and pitching motionsand the equation ofmotion for the floating breakwater systemare formulated and calculated for various case scenariosby varying the wave conditions and porous effect param-eters Considering the limiting case without porous wallsgood agreements between the present analytical solutionsand published numerical results are achieved This suggeststhat the present analytical model is demonstrated to be areliable engineering tool to provide reasonable estimationsfor the hydrodynamic performances of a floating breakwatersystem with dual side porous walls It is worthy to noticethat the wave loads on the floating structures especiallythe moments when referenced to the rotational center ofbottom center would be reduced with an increase in theporous-effect parameters of the porous walls Additionallythe existence of the dual porous walls could move the peaksof surge and pitching RAOs to the waves of higher frequencywhen the porous effect parameters decrease Overall it canbe concluded that a properly adopted porous-wall systemwhen attached to a floating body can effectively improveits hydrodynamic performance as a breakwater The analyt-ical approach developed in this study can also be furtherextended to assist the theoretical analysis for other similar butimproved floating breakwater systems
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was supported by ldquoFundamental Research Fundsfor the Central Universitiesrdquo (Grand No 017180224)
References
[1] I-H Cho ldquoTransmission coefficients of a floating rectangularbreakwater with porous side platesrdquo International Journal ofNaval Architecture and Ocean Engineering vol 8 no 1 pp 53ndash65 2016
[2] H Hu K-H Wang and A N Williams ldquoWave motion over abreakwater system of a horizontal plate and a vertical porouswallrdquo Ocean Engineering vol 29 no 4 pp 373ndash386 2001
[3] C C Mei and J L Black ldquoScattering of surface waves byrectangular obstacles in water of finite depthrdquo Journal of FluidMechanics vol 38 no 3 pp 499ndash511 1969
[4] J L Black C C Mei and M C G Bray ldquoRadiation andscattering of water waves by rigid bodiesrdquo Journal of FluidMechanics vol 46 no 1 pp 151ndash164 1971
[5] N Drimer Y Agnon and M Stiassnie ldquoA simplified analyticalmodel for a floating breakwater in water of finite depthrdquoAppliedOcean Research vol 14 no 1 pp 33ndash41 1992
[6] J-F Lee ldquoOn the heave radiation of a rectangular structurerdquoOcean Engineering vol 22 no 1 pp 19ndash34 1995
[7] Y H Zheng Y G You and Y M Shen ldquoOn the radiationand diffraction of water waves by a rectangular buoyrdquo OceanEngineering vol 31 no 8-9 pp 1063ndash1082 2004
[8] E D Christensen H B Bingham A P Skou Friis A K Larsenand K L Jensen ldquoAn experimental and numerical study offloating breakwatersrdquo Coastal Engineering Journal vol 137 pp43ndash58 2018
[9] E Koutandos P Prinos andXGironella ldquoFloating breakwatersunder regular and irregular wave forcing Reflection and trans-mission characteristicsrdquo Journal of Hydraulic Research vol 43no 2 pp 174ndash188 2005
[10] G H Dong Y N Zheng Y C Li B Teng C T Guan and D FLin ldquoExperiments on wave transmission coefficients of floatingbreakwatersrdquo Ocean Engineering vol 35 no 8-9 pp 931ndash9382008
[11] A T Chwang ldquoA porous-wavemaker theoryrdquo Journal of FluidMechanics vol 132 pp 395ndash406 1983
[12] Y C Li Y Liu and B Teng ldquoPorous effect parameter of thinpermeable platesrdquo Coastal Engineering Journal vol 48 no 4pp 309ndash336 2006
[13] S R Manam and M Sivanesan ldquoScattering of water wavesby vertical porous barriers an analytical approachrdquo WaveMotion An International Journal Reporting Research on WavePhenomena vol 67 pp 89ndash101 2016
[14] A T Chwang and Z Dong ldquoWave-trapping due to a porousplaterdquo in Proceedings of 15 ONR symposium on Naval Hydrody-namics pp 407ndash414 1985
[15] S Zhu and A T Chwang ldquoAnalytical study of porous waveabsorberrdquo Journal of Engineering Mechanics vol 127 no 4 pp326ndash332 2001
[16] H Behera and T Sahoo ldquoHydroelastic analysis of gravity waveinteraction with submerged horizontal flexible porous platerdquoJournal of Fluids and Structures vol 54 pp 643ndash660 2015
[17] T L Yip and A T Chwang ldquoPerforated wall breakwater withinternal horizontal platerdquo Journal of Engineering Mechanics vol126 no 5 pp 533ndash538 2000
[18] Y Liu and H-J Li ldquoWave scattering by dual submergedhorizontal porous plates Further resultsrdquo Ocean Engineeringvol 81 no 5 pp 158ndash163 2014
[19] I H Cho and M H Kim ldquoWave absorbing system usinginclined perforated platesrdquo Journal of Fluid Mechanics vol 608pp 1ndash20 2008
[20] K-H Wang and X Ren ldquoWater waves on flexible and porousbreakwatersrdquo Journal of Engineering Mechanics vol 119 no 5pp 1025ndash1047 1993
[21] K H Wang and X Ren ldquoWave interaction with a concentricporous cylinder systemrdquo Ocean Engineering vol 21 no 4 pp343ndash360 1994
[22] F Zhao T Zhang R Wan L Huang X Wang and WBao ldquoHydrodynamic loads acting on a circular porous platehorizontally submerged in wavesrdquo Ocean Engineering vol 136pp 168ndash177 2017
14 Mathematical Problems in Engineering
[23] A Mondal and R Gayen ldquoWave interaction with dual circularporous platesrdquo Journal of Marine Science and Application vol14 no 4 pp 366ndash375 2015
[24] W L Qiao K-H Wang and Y Q Sun ldquoScattering of waterwaves by a floating body with two vertically attached porouswallsrdquo Journal of Engineering Mechanics vol 144 no 2 ArticleID 04017162 2018
[25] S W Twu and D T Lin ldquoOn a highly effective wave absorberrdquoCoastal Engineering Journal vol 15 no 4 pp 389ndash405 1991
[26] X P Yu ldquoDiffraction of water waves by porous breakwatersrdquoJournal of Waterway Port Coastal and Ocean Engineering vol121 no 6 pp 275ndash282 1995
[27] I J Losada M A Losada and A Baquerizo ldquoAn analyticalmethod to evaluate the efficiency of porous screens as wavedampersrdquo Applied Ocean Research vol 15 no 4 pp 207ndash2151993
[28] G Besio and M A Losada ldquoSediment transport patterns atTrafalgar offshore windfarmrdquo Ocean Engineering vol 35 no 7pp 653ndash665 2008
[29] H Song and L Tao ldquoShort-crestedwave interactionwith a con-centric porous cylindrical structurerdquo Applied Ocean Researchvol 29 no 4 pp 199ndash209 2007
[30] L Xiao Y Kou L Tao and L Yang ldquoComparative studyof hydrodynamic performances of breakwaters with double-layered perforated walls attached to ring-shaped very largefloating structuresrdquo Ocean Engineering vol 111 pp 279ndash2912016
[31] T L Yip and A T Chwang ldquoWater wave control by submergedpitching porous platerdquo Journal of Engineering Mechanics vol124 no 4 pp 428ndash434 1998
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Submit your manuscripts atwwwhindawicom
8 Mathematical Problems in Engineering
119871119896 can be computed with formulations given below
11 = (int0minus119889
120601(1)1198771 10038161003816100381610038161003816119909=minus119886 119889119911 minus int0minus119889
120601(1)1198773 10038161003816100381610038161003816119909=119886 119889119911)+ (intminus119904minus119889
120601(1)1198772 10038161003816100381610038161003816119909=minus119886 119889119911 minus intminus119904minus119889
120601(1)1198772 10038161003816100381610038161003816119909=119886 119889119911)= 119872sum119898=1
119860(1)119898 minus 119861(1)119898119898 cos 119898ℎ [sin 119898ℎ minus sin 119898 (ℎ minus 119889)]+ 2119886119862(1)0 (119889 minus 119904) + 119872sum
119898=1
2119864(1)119898119903119898 [sin119898120587minus sin 119903119898 (ℎ minus 119889)]
(58)
22 = int119886minus119886
120601(2)1198772 10038161003816100381610038161003816119911=minus119909 119889119909 = 2119886119864(2)0+ 119872sum119898=1
2119862(2)119898119903119898 tanh (119903119898119886) cos 119903119898 (ℎ minus 119904) + 119886 (ℎ minus 119904)minus 11988633 (ℎ minus 119904)
(59)
33 = int0minus119889
( 120601(3)1198771 10038161003816100381610038161003816119909=minus119886 minus 120601(3)1198773 10038161003816100381610038161003816119909=119886) (119911 minus 120585119911) 119889119911+ intminus119904minus119889
(120601(3)1198772 10038161003816100381610038161003816119909=minus119886 minus 120601(3)1198772 10038161003816100381610038161003816119909=119886) (119911 minus 120585119911) 119889119911minus int119886minus119886
120601(3)1198772 10038161003816100381610038161003816119909=minus119904 119909 119889119909 = 119872sum119898=0
(119860(3)119898 minus 119861(3)119898 )
sdot [119889 sin 119898 (ℎ minus 119889) minus 120585119911 sin 119898 (ℎ minus 119889)119898 cos 119898ℎ
+ cos 119898ℎ minus cos 119898 (ℎ minus 119889)1198982 cos 119898ℎ ] + 119886119862(3)0 (1198892 minus 1199042)
minus 119872sum119898=1
2119864(3)119898119903119898 [119889 sin 119898 (ℎ minus 119889)+ cos 119903119898 (ℎ minus 119904) minus cos 119903119898 (ℎ minus 119889)119903119898 ] + 2119886119862(3)0 120585119911 (119889minus 119904) minus 119872sum
119898=1
2119864(3)119898 120585119911119903119898 sin 119903119898 (ℎ minus 119889) minus 23 119862(3)0 1198863 minus 119872sum119898=1
120576sdot cosh 120574119898 (ℎ minus 119904) 1 minus cos 21198861205741198981205741198982 + 119872sum
119898=1
(120576 cos119898120587minus 120576) [ 119889 sinh 120574119898 (ℎ minus 119889) minus 119904 sinh 120574119898 (ℎ minus 119904)120574119898minus cosh 120574119898 (ℎ minus 119904) minus cosh 120574119898 (ℎ minus 119889)1205741198982 ]minus 119872sum119898=1
119864(3)119898 cos 119903119898 (h minus s)119903119898 sinh 119903119898119886 (2119886 cosh 119903119898119886minus 2 sinh 119903119898119886119903119898 )
(60)
31 = 13 = int0minus119889
( 120601(3)1198771 10038161003816100381610038161003816119909=minus119886 minus 120601(3)1198773 10038161003816100381610038161003816119909=119886) 119889119911+ intminus119904minus119889
( 120601(3)1198772 10038161003816100381610038161003816119909=minus119886 minus 120601(3)1198772 10038161003816100381610038161003816119909=119886) 119889119911 = 119872sum119898=0
(119860(3)119898minus 119861(3)119898 ) sin 119898ℎ minus sin 119898 (ℎ minus 119889)
119898 cos 119898ℎ minus 2 (ℎ minus 119904)sdot [ 11988636 (ℎ minus 119904) + 119886119862(3)0 ] + 119886 (ℎ minus 119904)3 minus 119886 (ℎ minus 119889)33 (ℎ minus 119904)+ 119872sum119898=1
2119864(3)119898119903119898 sin 119903119898 (ℎ minus 119889)
(61)
33 Equation of Motion and RAOs The derived solutions forthe diffraction and radiation problems could now be inte-grated with the equation of motion of the floating breakwatersystem to analyze the dynamic responses of the structuralsystem We have
minus1205962 [119872 + 119898] minus i120596 [119889] + [119904] [1198830] = [1198650] (62)
where [1198830] = [11988301 11988302 11988303]119879 and are vectors indicatingthe motion amplitudes and the wave exciting forces respec-tively [119898] and [119889] are addedmass and dampingmatrices [119872]is the mass matrix and [119878] denotes the hydrostatic restoringforce matrix they can be expressed as
[119872] = [[[
1198980 0 minus11989801198851198660 1198980 0minus1198980119885119866 0 1]]]
(63)
[119878] = [[[
0 0 00 2120588119892119886 00 0 1198980119892119866119872
]]]
(64)
Here 1198980 is the mass per unit length of the floatingbreakwater and 119911119866 is the z-coordinate of the center of gravityThe mass moment of inertia about y-axis I can be computedby 119868 = 11989801199031198662 in which 119903119866 is the radius of gyration in pitchingoscillation and 119866119872 is the metacentric height of the floatingbreakwater system Once 1198830119895 (j=1 2 3) are determined bysolving (62) the RAOs (Response Amplitude Operators) ofthe breakwater system can be calculated as
119877119860119874119895 = 1198830119895119867119895 (119895 = 1 2 3) (65)
4 Results and Discussion
The validation of the wave diffraction solutions in terms ofreflectiontransmission coefficients and hydrodynamic loadsfor a train of monochromatic wave propagating throughthe proposed floating breakwater system that consists of apartially submerged rectangular body and with or withoutthe attached two porous side walls has been carried outin a previous study (Qiao et al [24]) Here additional
Mathematical Problems in Engineering 9
Black et al(1971)Present approach
G=G=infinhs=as=
0010203040506070809
F D
gH
a
3 60 5 71 42ks
(a) Horizontal force
Black et al(1971)Present approach
G=G=infinhs= as=
0010203040506070809
1
F D
gHa
3 60 5 71 42ks
(b) Vertical force
Black et al(1971)Present approach
G=G=infinhs= as=
3 60 4 51 72ks
0
005
01
015
02
025
F D
gH
a2
(c) Moment
Figure 2 Comparison of the dimensionless wave exciting forces and moment versus 1198960119904 for ℎ119904 = 20 119886119904 = 30 11986601 = 11986602 997888rarr infin (a)dimensionless horizontal exciting force (b) dimensionless vertical exciting force (c) dimensionless exciting moment about the rotationalcenter of (0 0)
comparisons of wave diffraction results with other publishednumerical solutions for the limiting case of neglecting theeffect of two porous walls are again presented For thewave radiation problem the motion induced hydrodynamiccoefficients such as added mass and damping coefficient arealso investigated for their variations with changes of waveparameters As a validation those results for the caseswithoutporous walls are compared with limited numerical solutionspublished in literatures Subsequently the effects of variousdimensionless parameters for example porous propertiesincident wave conditions and the geometry of floating bodyto wave-induced hydrodynamic forces hydrodynamic coeffi-cients and RAOs were also examined In our computationsthe first 40 terms in the infinite series of the diffractionpotentials and radiation potentials are taken the same as [7]namely M=40
As an example showing the results comparisons for thediffraction problem Figure 2 illustrates the variations ofthe dimensionless exciting forces in horizontal and verticaldirections versus dimensionless wavelength 1198960119904 for the caseof ℎ119904 = 20 119886119904 = 10 11986601 = 11986602 997888rarr infin The excitingmoments with reference to the rotational center of (00) are
also presented in Figure 2 Here 11986601 = 11986602 997888rarr infin representsthe case of a floating breakwater system without dual porousside walls The numerical results computed by Black et al [4]are also plotted in Figure 2 The comparisons indicate thatthe exciting forces and moments are well predicted by thepresent analytical model Both the analytical solutions andthe numerical results reveal that the horizontal exciting forcesand the exciting moments show rapidly increasing trendinitially until a peak value is reached and then are followedby a gradual decrease of the values with the further increaseof 1198960119904 However the vertical forces as shown in Figure 2(b)decrease continuously with an increase in 1198960119904 According tothe results given in the previous study by Qiao et al [24] itindicated that the effectiveness of the porous walls is reflectedwith the reduction of the vertical hydrodynamic forces andwhen the breakwater system is rotated with reference to thebottom center of the body the induced moment can also bereduced However due to the added additional length of theporous walls the horizontal forces are slightly greater thanthose without porous walls
Figure 3 presents the variations of the dimensionlessadded mass and damping coefficients in surge only motion
10 Mathematical Problems in Engineering
YHZheng et al(2004)
G=G=
G=G=
G=G=
G=G=infin
0
1
2
3
4
5
6m
2
as
963 100 87521 114kh
(a) Added mass
YH Zheng et al(2004)
G=G=infin
G=G=
G=G=
G=G=
963 100 87521 114kh
005
115
225
335
4
d
2
as
(b) Radiation coefficient
Figure 3 Variation of the dimensionless surge-surge hydrodynamic coefficients ((a) for the added mass and (b) for the radiation damping)versus 1198960ℎ under the conditions of different porous-effect parameters for 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16
YHZheng et al(2004)
G=G=
G=G=
G=G=
G=G=infin
963 100 87521 114kh
0
02
04
06
08
1
12
14
m
2
as
(a) Added mass
YHZheng et al(2004)
G=G=
G=G=
G=G=
G=G=infin
minus04minus02
002040608
11214
d
2
as
963 100 87521 114kh
(b) Radiation damping
Figure 4 Variation of the dimensionless heave-heave hydrodynamic coefficients ((a) for the added mass and (b) for the radiation damping)versus 1198960ℎ under the conditions of different porous-effect parameters for 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16
versus 1198960ℎ for the setting of 119904ℎ = 13 119889ℎ = 12 and 119886ℎ =16under the conditions of various porous-effect parametersWhen 11986601 = 11986602 = infin it represents the case withoutconsidering the effect of dual porous walls The presentanalytical solutions for this limiting case can be adopted tocompare to the limited numerical results available in thestudy of [7] The comparison plot is shown in Figure 4 wherethe nearly identical matches between the present analyticalsolutions and the numerical results of [7] can be observedThe results shown in Figure 4 suggest that the addition ofporous walls tends to enhance the added mass and dampingcoefficients In general with a decrease of porous effectparameter the addedmass and damping coefficients increaseFor the variation of addedmass coefficient with porouswalls
it can be seen that a peak value occurred at a long wave (small1198960ℎ value) condition However for the radiation dampingcoefficient it is shown to increase rapidly to its maximumvalue with an increase of 1198960ℎ then a gradual decrease trendis followed by a further increase of 1198960ℎ
The dimensionless heave-heave added mass (119898222120588119886119904)and radiation damping coefficient (119889222120588120596119886119904) versus 1198960ℎunder various 11986601 and 11986602 values for the case of 119904ℎ =13 119889ℎ = 12 119886ℎ = 16 are presented in Figure 4Under the limiting condition of 11986601 = 11986602 = infin (withoutporous walls) the numerical results produced by the study of[7] are also plotted for comparison Again good agreementsbetween the present analytical solutions and the publishednumerical results can be noticed The results in Figure 4(a)
Mathematical Problems in Engineering 11
YHZheng et al(2004)
963 100 87521 114kh
0
02
04
06
08
1
12
14m
2
as
G=G=infin
G=G=
G=G=
G=G=
(a) Added mass
YH Zhang et al(2004)
G=G=infin
G=G=
G=G=
G=G=
0010203040506070809
d
2
as
963 100 87521 114kh
(b) Radiation damping
Figure 5 Variation of the dimensionless pitch-pitch hydrodynamic coefficients ((a) for the added mass and (b) for the radiation damping)versus 1198960ℎ under the conditions of different porous-effect parameters for 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16 with the rotational center set at (00)
suggest that the variations of added mass for the shorterwavelength waves (large 1198960ℎ values) are less sensitive to thechange of porous parameters The presence of dual porouswalls increases the heave-heave added mass which tends toreach its maximum value at the long wave extreme Alsoa decrease in porous parameters results in an increase inheave induced added mass As shown in Figure 4(b) theheave-heave damping coefficients decrease rapidly when 1198960ℎincreases initially Then a gradual changing trend is followedby a further increase in 1198960ℎ Additionally the presence ofdual porous walls tends to reduce the values of heave-heavedamping coefficients
For the pitching induced added mass and dampingcoefficients the results are plotted against 1198960ℎ in Figure 5for various porous-effect parameters 11986601 and 11986602 and underthe settings of 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16 Whenthe rotational center is set at (00) as shown in Figure 5 thepresent analytical solutions match well with the publishednumerical results [7] for the limiting case of no porouswall (ie 11986601 = 11986602 = infin) It can be observed thatthe pitching oscillation induced added mass and dampingcoefficients for the floating breakwater system with dualporous side walls are greater than those without porous wallsWith decrease of the porous effect parameters the addedmass and damping coefficients increase For the added masscoefficients it is found that the variations trends are morecomplex for waves with relatively smaller 1198960ℎ under theconditions of different porous-effect parameters Generallyin cases of weaker porous-effect (for example 11986601 = 11986602 le 2as shown in Figure 5(a)) the value of added mass coefficienttends to approach 14 under the case considered For thevariations of damping coefficient as shown in Figure 5(b) thevarying trends are similar to those of damping coefficient insurge motion
The results presented in Figure 6 are the variations ofthe dimensionless surge-pitch added mass and damping
coefficients versus 1198960ℎ for varying 11986601 and 11986602 The case isfor 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16 with referenceto the rotational center of (00) Clearly the added massand damping coefficients are shown to increase with anincrease in the porous-effect parameters of both porouswalls A and B while the damping coefficients reach theirabsolute maximum values when 1198960ℎ is within the rangeof 1 and 2 The magnitude of surge-pitch hydrodynamiccoefficients especially the damping is greatly influenced bythe permeability of the dual porous walls compared to thesituation with surge or pitching motion only
The analytical solutions for the Response AmplitudeOperators (RAOs) of the floating breakwater system pro-posed in this study are illustrated in Figure 7 as a functionof dimensionless wavelength 1198960ℎ for various porous-effectparameters with the conditions of 119904ℎ = 03 119889ℎ = 05119886ℎ = 025 119903119866119886 = 09 and 119866119872119886 = 01 where therotational center is located at (00) It is found that the varyingtrends of surge RAOs are similar to those of pitching RAOsalthough the peak values of surge RAOs are smaller thanthose of pitching RAOs When the porous-effect parametersdecrease the main peak values of either surge or pitchingRAOs that occurred are shifted to the wave conditions withshorter wavelength Furthermore for the heave RAO itexperiences a typical damped response for the case of singledegree of freedom system and its maximum values of RAOwould move to the lower frequency wave region when theporous-effect parameters decrease
5 Summary and Conclusions
The hydrodynamic performance investigations includingthe hydrodynamic loads radiation induced added massand damping coefficients and dynamic responses relatedRAOs are critical to the study that is associated with theproblems of wave and structure interactions In this study the
12 Mathematical Problems in Engineering
G=G=infin
G=G=
G=G=
G=G=
963 100 87521 114kh
minus4
minus3
minus2
minus1
0
1
m
2
as
(a) Added mass
G=G=infin
G=G=
G=G=
G=G=
963 100 87521 114kh
minus3
minus25
minus2
minus15
minus1
minus05
0
d
2
as
(b) Radiation damping
Figure 6 Variation of the dimensionless surge-pitch hydrodynamic coefficients ((a) for the added mass and (b) for the radiation damping)versus 1198960ℎ under the conditions of different porous-effect parameters for 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16 with the rotational center set at(0 0)
G=G=
G=G=
G=G=
G=G=infin
4515 3525 3 605 2 5 551 4kh
0123456789
1011
|X|A
(a) Surge RAO
G=G=infin
G=G=
G=G=
G=G=
3 60 4 51 2kh
0
05
1
15
2
25
3
35|X
|A
(b) Heave RAO
G=G=
G=G=
G=G=
G=G=infin
05 3515 325 45 65 5541 20kh
0
5
10
15
|X|A
(c) Pitch RAO
Figure 7 Variation of the dimensionless RAO ((a) for surge RAO (b) for heave RAO and (c) for pitch RAO) versus 1198960ℎ under the conditionsof different porous-effect parameters for 119904ℎ = 03 119889ℎ = 05 119886ℎ = 025 119903119866119886 = 09 119866119872119886 = 01 with rotation center at (00)
Mathematical Problems in Engineering 13
hydrodynamic problems considering a floating breakwatersystem that consists of a rectangular body and two verticalattached porous walls are investigated analytically based onthe two-dimensional potential flow theory The expressionsof the diffraction and radiation based velocity potentials andthe associated unknown coefficients for each subdomain arederived by the MEEM approach Subsequently the excitinghydrodynamic forces added mass and radiation dampingcoefficients in terms of surge heave and pitching motionsand the equation ofmotion for the floating breakwater systemare formulated and calculated for various case scenariosby varying the wave conditions and porous effect param-eters Considering the limiting case without porous wallsgood agreements between the present analytical solutionsand published numerical results are achieved This suggeststhat the present analytical model is demonstrated to be areliable engineering tool to provide reasonable estimationsfor the hydrodynamic performances of a floating breakwatersystem with dual side porous walls It is worthy to noticethat the wave loads on the floating structures especiallythe moments when referenced to the rotational center ofbottom center would be reduced with an increase in theporous-effect parameters of the porous walls Additionallythe existence of the dual porous walls could move the peaksof surge and pitching RAOs to the waves of higher frequencywhen the porous effect parameters decrease Overall it canbe concluded that a properly adopted porous-wall systemwhen attached to a floating body can effectively improveits hydrodynamic performance as a breakwater The analyt-ical approach developed in this study can also be furtherextended to assist the theoretical analysis for other similar butimproved floating breakwater systems
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was supported by ldquoFundamental Research Fundsfor the Central Universitiesrdquo (Grand No 017180224)
References
[1] I-H Cho ldquoTransmission coefficients of a floating rectangularbreakwater with porous side platesrdquo International Journal ofNaval Architecture and Ocean Engineering vol 8 no 1 pp 53ndash65 2016
[2] H Hu K-H Wang and A N Williams ldquoWave motion over abreakwater system of a horizontal plate and a vertical porouswallrdquo Ocean Engineering vol 29 no 4 pp 373ndash386 2001
[3] C C Mei and J L Black ldquoScattering of surface waves byrectangular obstacles in water of finite depthrdquo Journal of FluidMechanics vol 38 no 3 pp 499ndash511 1969
[4] J L Black C C Mei and M C G Bray ldquoRadiation andscattering of water waves by rigid bodiesrdquo Journal of FluidMechanics vol 46 no 1 pp 151ndash164 1971
[5] N Drimer Y Agnon and M Stiassnie ldquoA simplified analyticalmodel for a floating breakwater in water of finite depthrdquoAppliedOcean Research vol 14 no 1 pp 33ndash41 1992
[6] J-F Lee ldquoOn the heave radiation of a rectangular structurerdquoOcean Engineering vol 22 no 1 pp 19ndash34 1995
[7] Y H Zheng Y G You and Y M Shen ldquoOn the radiationand diffraction of water waves by a rectangular buoyrdquo OceanEngineering vol 31 no 8-9 pp 1063ndash1082 2004
[8] E D Christensen H B Bingham A P Skou Friis A K Larsenand K L Jensen ldquoAn experimental and numerical study offloating breakwatersrdquo Coastal Engineering Journal vol 137 pp43ndash58 2018
[9] E Koutandos P Prinos andXGironella ldquoFloating breakwatersunder regular and irregular wave forcing Reflection and trans-mission characteristicsrdquo Journal of Hydraulic Research vol 43no 2 pp 174ndash188 2005
[10] G H Dong Y N Zheng Y C Li B Teng C T Guan and D FLin ldquoExperiments on wave transmission coefficients of floatingbreakwatersrdquo Ocean Engineering vol 35 no 8-9 pp 931ndash9382008
[11] A T Chwang ldquoA porous-wavemaker theoryrdquo Journal of FluidMechanics vol 132 pp 395ndash406 1983
[12] Y C Li Y Liu and B Teng ldquoPorous effect parameter of thinpermeable platesrdquo Coastal Engineering Journal vol 48 no 4pp 309ndash336 2006
[13] S R Manam and M Sivanesan ldquoScattering of water wavesby vertical porous barriers an analytical approachrdquo WaveMotion An International Journal Reporting Research on WavePhenomena vol 67 pp 89ndash101 2016
[14] A T Chwang and Z Dong ldquoWave-trapping due to a porousplaterdquo in Proceedings of 15 ONR symposium on Naval Hydrody-namics pp 407ndash414 1985
[15] S Zhu and A T Chwang ldquoAnalytical study of porous waveabsorberrdquo Journal of Engineering Mechanics vol 127 no 4 pp326ndash332 2001
[16] H Behera and T Sahoo ldquoHydroelastic analysis of gravity waveinteraction with submerged horizontal flexible porous platerdquoJournal of Fluids and Structures vol 54 pp 643ndash660 2015
[17] T L Yip and A T Chwang ldquoPerforated wall breakwater withinternal horizontal platerdquo Journal of Engineering Mechanics vol126 no 5 pp 533ndash538 2000
[18] Y Liu and H-J Li ldquoWave scattering by dual submergedhorizontal porous plates Further resultsrdquo Ocean Engineeringvol 81 no 5 pp 158ndash163 2014
[19] I H Cho and M H Kim ldquoWave absorbing system usinginclined perforated platesrdquo Journal of Fluid Mechanics vol 608pp 1ndash20 2008
[20] K-H Wang and X Ren ldquoWater waves on flexible and porousbreakwatersrdquo Journal of Engineering Mechanics vol 119 no 5pp 1025ndash1047 1993
[21] K H Wang and X Ren ldquoWave interaction with a concentricporous cylinder systemrdquo Ocean Engineering vol 21 no 4 pp343ndash360 1994
[22] F Zhao T Zhang R Wan L Huang X Wang and WBao ldquoHydrodynamic loads acting on a circular porous platehorizontally submerged in wavesrdquo Ocean Engineering vol 136pp 168ndash177 2017
14 Mathematical Problems in Engineering
[23] A Mondal and R Gayen ldquoWave interaction with dual circularporous platesrdquo Journal of Marine Science and Application vol14 no 4 pp 366ndash375 2015
[24] W L Qiao K-H Wang and Y Q Sun ldquoScattering of waterwaves by a floating body with two vertically attached porouswallsrdquo Journal of Engineering Mechanics vol 144 no 2 ArticleID 04017162 2018
[25] S W Twu and D T Lin ldquoOn a highly effective wave absorberrdquoCoastal Engineering Journal vol 15 no 4 pp 389ndash405 1991
[26] X P Yu ldquoDiffraction of water waves by porous breakwatersrdquoJournal of Waterway Port Coastal and Ocean Engineering vol121 no 6 pp 275ndash282 1995
[27] I J Losada M A Losada and A Baquerizo ldquoAn analyticalmethod to evaluate the efficiency of porous screens as wavedampersrdquo Applied Ocean Research vol 15 no 4 pp 207ndash2151993
[28] G Besio and M A Losada ldquoSediment transport patterns atTrafalgar offshore windfarmrdquo Ocean Engineering vol 35 no 7pp 653ndash665 2008
[29] H Song and L Tao ldquoShort-crestedwave interactionwith a con-centric porous cylindrical structurerdquo Applied Ocean Researchvol 29 no 4 pp 199ndash209 2007
[30] L Xiao Y Kou L Tao and L Yang ldquoComparative studyof hydrodynamic performances of breakwaters with double-layered perforated walls attached to ring-shaped very largefloating structuresrdquo Ocean Engineering vol 111 pp 279ndash2912016
[31] T L Yip and A T Chwang ldquoWater wave control by submergedpitching porous platerdquo Journal of Engineering Mechanics vol124 no 4 pp 428ndash434 1998
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Mathematical Problems in Engineering 9
Black et al(1971)Present approach
G=G=infinhs=as=
0010203040506070809
F D
gH
a
3 60 5 71 42ks
(a) Horizontal force
Black et al(1971)Present approach
G=G=infinhs= as=
0010203040506070809
1
F D
gHa
3 60 5 71 42ks
(b) Vertical force
Black et al(1971)Present approach
G=G=infinhs= as=
3 60 4 51 72ks
0
005
01
015
02
025
F D
gH
a2
(c) Moment
Figure 2 Comparison of the dimensionless wave exciting forces and moment versus 1198960119904 for ℎ119904 = 20 119886119904 = 30 11986601 = 11986602 997888rarr infin (a)dimensionless horizontal exciting force (b) dimensionless vertical exciting force (c) dimensionless exciting moment about the rotationalcenter of (0 0)
comparisons of wave diffraction results with other publishednumerical solutions for the limiting case of neglecting theeffect of two porous walls are again presented For thewave radiation problem the motion induced hydrodynamiccoefficients such as added mass and damping coefficient arealso investigated for their variations with changes of waveparameters As a validation those results for the caseswithoutporous walls are compared with limited numerical solutionspublished in literatures Subsequently the effects of variousdimensionless parameters for example porous propertiesincident wave conditions and the geometry of floating bodyto wave-induced hydrodynamic forces hydrodynamic coeffi-cients and RAOs were also examined In our computationsthe first 40 terms in the infinite series of the diffractionpotentials and radiation potentials are taken the same as [7]namely M=40
As an example showing the results comparisons for thediffraction problem Figure 2 illustrates the variations ofthe dimensionless exciting forces in horizontal and verticaldirections versus dimensionless wavelength 1198960119904 for the caseof ℎ119904 = 20 119886119904 = 10 11986601 = 11986602 997888rarr infin The excitingmoments with reference to the rotational center of (00) are
also presented in Figure 2 Here 11986601 = 11986602 997888rarr infin representsthe case of a floating breakwater system without dual porousside walls The numerical results computed by Black et al [4]are also plotted in Figure 2 The comparisons indicate thatthe exciting forces and moments are well predicted by thepresent analytical model Both the analytical solutions andthe numerical results reveal that the horizontal exciting forcesand the exciting moments show rapidly increasing trendinitially until a peak value is reached and then are followedby a gradual decrease of the values with the further increaseof 1198960119904 However the vertical forces as shown in Figure 2(b)decrease continuously with an increase in 1198960119904 According tothe results given in the previous study by Qiao et al [24] itindicated that the effectiveness of the porous walls is reflectedwith the reduction of the vertical hydrodynamic forces andwhen the breakwater system is rotated with reference to thebottom center of the body the induced moment can also bereduced However due to the added additional length of theporous walls the horizontal forces are slightly greater thanthose without porous walls
Figure 3 presents the variations of the dimensionlessadded mass and damping coefficients in surge only motion
10 Mathematical Problems in Engineering
YHZheng et al(2004)
G=G=
G=G=
G=G=
G=G=infin
0
1
2
3
4
5
6m
2
as
963 100 87521 114kh
(a) Added mass
YH Zheng et al(2004)
G=G=infin
G=G=
G=G=
G=G=
963 100 87521 114kh
005
115
225
335
4
d
2
as
(b) Radiation coefficient
Figure 3 Variation of the dimensionless surge-surge hydrodynamic coefficients ((a) for the added mass and (b) for the radiation damping)versus 1198960ℎ under the conditions of different porous-effect parameters for 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16
YHZheng et al(2004)
G=G=
G=G=
G=G=
G=G=infin
963 100 87521 114kh
0
02
04
06
08
1
12
14
m
2
as
(a) Added mass
YHZheng et al(2004)
G=G=
G=G=
G=G=
G=G=infin
minus04minus02
002040608
11214
d
2
as
963 100 87521 114kh
(b) Radiation damping
Figure 4 Variation of the dimensionless heave-heave hydrodynamic coefficients ((a) for the added mass and (b) for the radiation damping)versus 1198960ℎ under the conditions of different porous-effect parameters for 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16
versus 1198960ℎ for the setting of 119904ℎ = 13 119889ℎ = 12 and 119886ℎ =16under the conditions of various porous-effect parametersWhen 11986601 = 11986602 = infin it represents the case withoutconsidering the effect of dual porous walls The presentanalytical solutions for this limiting case can be adopted tocompare to the limited numerical results available in thestudy of [7] The comparison plot is shown in Figure 4 wherethe nearly identical matches between the present analyticalsolutions and the numerical results of [7] can be observedThe results shown in Figure 4 suggest that the addition ofporous walls tends to enhance the added mass and dampingcoefficients In general with a decrease of porous effectparameter the addedmass and damping coefficients increaseFor the variation of addedmass coefficient with porouswalls
it can be seen that a peak value occurred at a long wave (small1198960ℎ value) condition However for the radiation dampingcoefficient it is shown to increase rapidly to its maximumvalue with an increase of 1198960ℎ then a gradual decrease trendis followed by a further increase of 1198960ℎ
The dimensionless heave-heave added mass (119898222120588119886119904)and radiation damping coefficient (119889222120588120596119886119904) versus 1198960ℎunder various 11986601 and 11986602 values for the case of 119904ℎ =13 119889ℎ = 12 119886ℎ = 16 are presented in Figure 4Under the limiting condition of 11986601 = 11986602 = infin (withoutporous walls) the numerical results produced by the study of[7] are also plotted for comparison Again good agreementsbetween the present analytical solutions and the publishednumerical results can be noticed The results in Figure 4(a)
Mathematical Problems in Engineering 11
YHZheng et al(2004)
963 100 87521 114kh
0
02
04
06
08
1
12
14m
2
as
G=G=infin
G=G=
G=G=
G=G=
(a) Added mass
YH Zhang et al(2004)
G=G=infin
G=G=
G=G=
G=G=
0010203040506070809
d
2
as
963 100 87521 114kh
(b) Radiation damping
Figure 5 Variation of the dimensionless pitch-pitch hydrodynamic coefficients ((a) for the added mass and (b) for the radiation damping)versus 1198960ℎ under the conditions of different porous-effect parameters for 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16 with the rotational center set at (00)
suggest that the variations of added mass for the shorterwavelength waves (large 1198960ℎ values) are less sensitive to thechange of porous parameters The presence of dual porouswalls increases the heave-heave added mass which tends toreach its maximum value at the long wave extreme Alsoa decrease in porous parameters results in an increase inheave induced added mass As shown in Figure 4(b) theheave-heave damping coefficients decrease rapidly when 1198960ℎincreases initially Then a gradual changing trend is followedby a further increase in 1198960ℎ Additionally the presence ofdual porous walls tends to reduce the values of heave-heavedamping coefficients
For the pitching induced added mass and dampingcoefficients the results are plotted against 1198960ℎ in Figure 5for various porous-effect parameters 11986601 and 11986602 and underthe settings of 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16 Whenthe rotational center is set at (00) as shown in Figure 5 thepresent analytical solutions match well with the publishednumerical results [7] for the limiting case of no porouswall (ie 11986601 = 11986602 = infin) It can be observed thatthe pitching oscillation induced added mass and dampingcoefficients for the floating breakwater system with dualporous side walls are greater than those without porous wallsWith decrease of the porous effect parameters the addedmass and damping coefficients increase For the added masscoefficients it is found that the variations trends are morecomplex for waves with relatively smaller 1198960ℎ under theconditions of different porous-effect parameters Generallyin cases of weaker porous-effect (for example 11986601 = 11986602 le 2as shown in Figure 5(a)) the value of added mass coefficienttends to approach 14 under the case considered For thevariations of damping coefficient as shown in Figure 5(b) thevarying trends are similar to those of damping coefficient insurge motion
The results presented in Figure 6 are the variations ofthe dimensionless surge-pitch added mass and damping
coefficients versus 1198960ℎ for varying 11986601 and 11986602 The case isfor 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16 with referenceto the rotational center of (00) Clearly the added massand damping coefficients are shown to increase with anincrease in the porous-effect parameters of both porouswalls A and B while the damping coefficients reach theirabsolute maximum values when 1198960ℎ is within the rangeof 1 and 2 The magnitude of surge-pitch hydrodynamiccoefficients especially the damping is greatly influenced bythe permeability of the dual porous walls compared to thesituation with surge or pitching motion only
The analytical solutions for the Response AmplitudeOperators (RAOs) of the floating breakwater system pro-posed in this study are illustrated in Figure 7 as a functionof dimensionless wavelength 1198960ℎ for various porous-effectparameters with the conditions of 119904ℎ = 03 119889ℎ = 05119886ℎ = 025 119903119866119886 = 09 and 119866119872119886 = 01 where therotational center is located at (00) It is found that the varyingtrends of surge RAOs are similar to those of pitching RAOsalthough the peak values of surge RAOs are smaller thanthose of pitching RAOs When the porous-effect parametersdecrease the main peak values of either surge or pitchingRAOs that occurred are shifted to the wave conditions withshorter wavelength Furthermore for the heave RAO itexperiences a typical damped response for the case of singledegree of freedom system and its maximum values of RAOwould move to the lower frequency wave region when theporous-effect parameters decrease
5 Summary and Conclusions
The hydrodynamic performance investigations includingthe hydrodynamic loads radiation induced added massand damping coefficients and dynamic responses relatedRAOs are critical to the study that is associated with theproblems of wave and structure interactions In this study the
12 Mathematical Problems in Engineering
G=G=infin
G=G=
G=G=
G=G=
963 100 87521 114kh
minus4
minus3
minus2
minus1
0
1
m
2
as
(a) Added mass
G=G=infin
G=G=
G=G=
G=G=
963 100 87521 114kh
minus3
minus25
minus2
minus15
minus1
minus05
0
d
2
as
(b) Radiation damping
Figure 6 Variation of the dimensionless surge-pitch hydrodynamic coefficients ((a) for the added mass and (b) for the radiation damping)versus 1198960ℎ under the conditions of different porous-effect parameters for 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16 with the rotational center set at(0 0)
G=G=
G=G=
G=G=
G=G=infin
4515 3525 3 605 2 5 551 4kh
0123456789
1011
|X|A
(a) Surge RAO
G=G=infin
G=G=
G=G=
G=G=
3 60 4 51 2kh
0
05
1
15
2
25
3
35|X
|A
(b) Heave RAO
G=G=
G=G=
G=G=
G=G=infin
05 3515 325 45 65 5541 20kh
0
5
10
15
|X|A
(c) Pitch RAO
Figure 7 Variation of the dimensionless RAO ((a) for surge RAO (b) for heave RAO and (c) for pitch RAO) versus 1198960ℎ under the conditionsof different porous-effect parameters for 119904ℎ = 03 119889ℎ = 05 119886ℎ = 025 119903119866119886 = 09 119866119872119886 = 01 with rotation center at (00)
Mathematical Problems in Engineering 13
hydrodynamic problems considering a floating breakwatersystem that consists of a rectangular body and two verticalattached porous walls are investigated analytically based onthe two-dimensional potential flow theory The expressionsof the diffraction and radiation based velocity potentials andthe associated unknown coefficients for each subdomain arederived by the MEEM approach Subsequently the excitinghydrodynamic forces added mass and radiation dampingcoefficients in terms of surge heave and pitching motionsand the equation ofmotion for the floating breakwater systemare formulated and calculated for various case scenariosby varying the wave conditions and porous effect param-eters Considering the limiting case without porous wallsgood agreements between the present analytical solutionsand published numerical results are achieved This suggeststhat the present analytical model is demonstrated to be areliable engineering tool to provide reasonable estimationsfor the hydrodynamic performances of a floating breakwatersystem with dual side porous walls It is worthy to noticethat the wave loads on the floating structures especiallythe moments when referenced to the rotational center ofbottom center would be reduced with an increase in theporous-effect parameters of the porous walls Additionallythe existence of the dual porous walls could move the peaksof surge and pitching RAOs to the waves of higher frequencywhen the porous effect parameters decrease Overall it canbe concluded that a properly adopted porous-wall systemwhen attached to a floating body can effectively improveits hydrodynamic performance as a breakwater The analyt-ical approach developed in this study can also be furtherextended to assist the theoretical analysis for other similar butimproved floating breakwater systems
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was supported by ldquoFundamental Research Fundsfor the Central Universitiesrdquo (Grand No 017180224)
References
[1] I-H Cho ldquoTransmission coefficients of a floating rectangularbreakwater with porous side platesrdquo International Journal ofNaval Architecture and Ocean Engineering vol 8 no 1 pp 53ndash65 2016
[2] H Hu K-H Wang and A N Williams ldquoWave motion over abreakwater system of a horizontal plate and a vertical porouswallrdquo Ocean Engineering vol 29 no 4 pp 373ndash386 2001
[3] C C Mei and J L Black ldquoScattering of surface waves byrectangular obstacles in water of finite depthrdquo Journal of FluidMechanics vol 38 no 3 pp 499ndash511 1969
[4] J L Black C C Mei and M C G Bray ldquoRadiation andscattering of water waves by rigid bodiesrdquo Journal of FluidMechanics vol 46 no 1 pp 151ndash164 1971
[5] N Drimer Y Agnon and M Stiassnie ldquoA simplified analyticalmodel for a floating breakwater in water of finite depthrdquoAppliedOcean Research vol 14 no 1 pp 33ndash41 1992
[6] J-F Lee ldquoOn the heave radiation of a rectangular structurerdquoOcean Engineering vol 22 no 1 pp 19ndash34 1995
[7] Y H Zheng Y G You and Y M Shen ldquoOn the radiationand diffraction of water waves by a rectangular buoyrdquo OceanEngineering vol 31 no 8-9 pp 1063ndash1082 2004
[8] E D Christensen H B Bingham A P Skou Friis A K Larsenand K L Jensen ldquoAn experimental and numerical study offloating breakwatersrdquo Coastal Engineering Journal vol 137 pp43ndash58 2018
[9] E Koutandos P Prinos andXGironella ldquoFloating breakwatersunder regular and irregular wave forcing Reflection and trans-mission characteristicsrdquo Journal of Hydraulic Research vol 43no 2 pp 174ndash188 2005
[10] G H Dong Y N Zheng Y C Li B Teng C T Guan and D FLin ldquoExperiments on wave transmission coefficients of floatingbreakwatersrdquo Ocean Engineering vol 35 no 8-9 pp 931ndash9382008
[11] A T Chwang ldquoA porous-wavemaker theoryrdquo Journal of FluidMechanics vol 132 pp 395ndash406 1983
[12] Y C Li Y Liu and B Teng ldquoPorous effect parameter of thinpermeable platesrdquo Coastal Engineering Journal vol 48 no 4pp 309ndash336 2006
[13] S R Manam and M Sivanesan ldquoScattering of water wavesby vertical porous barriers an analytical approachrdquo WaveMotion An International Journal Reporting Research on WavePhenomena vol 67 pp 89ndash101 2016
[14] A T Chwang and Z Dong ldquoWave-trapping due to a porousplaterdquo in Proceedings of 15 ONR symposium on Naval Hydrody-namics pp 407ndash414 1985
[15] S Zhu and A T Chwang ldquoAnalytical study of porous waveabsorberrdquo Journal of Engineering Mechanics vol 127 no 4 pp326ndash332 2001
[16] H Behera and T Sahoo ldquoHydroelastic analysis of gravity waveinteraction with submerged horizontal flexible porous platerdquoJournal of Fluids and Structures vol 54 pp 643ndash660 2015
[17] T L Yip and A T Chwang ldquoPerforated wall breakwater withinternal horizontal platerdquo Journal of Engineering Mechanics vol126 no 5 pp 533ndash538 2000
[18] Y Liu and H-J Li ldquoWave scattering by dual submergedhorizontal porous plates Further resultsrdquo Ocean Engineeringvol 81 no 5 pp 158ndash163 2014
[19] I H Cho and M H Kim ldquoWave absorbing system usinginclined perforated platesrdquo Journal of Fluid Mechanics vol 608pp 1ndash20 2008
[20] K-H Wang and X Ren ldquoWater waves on flexible and porousbreakwatersrdquo Journal of Engineering Mechanics vol 119 no 5pp 1025ndash1047 1993
[21] K H Wang and X Ren ldquoWave interaction with a concentricporous cylinder systemrdquo Ocean Engineering vol 21 no 4 pp343ndash360 1994
[22] F Zhao T Zhang R Wan L Huang X Wang and WBao ldquoHydrodynamic loads acting on a circular porous platehorizontally submerged in wavesrdquo Ocean Engineering vol 136pp 168ndash177 2017
14 Mathematical Problems in Engineering
[23] A Mondal and R Gayen ldquoWave interaction with dual circularporous platesrdquo Journal of Marine Science and Application vol14 no 4 pp 366ndash375 2015
[24] W L Qiao K-H Wang and Y Q Sun ldquoScattering of waterwaves by a floating body with two vertically attached porouswallsrdquo Journal of Engineering Mechanics vol 144 no 2 ArticleID 04017162 2018
[25] S W Twu and D T Lin ldquoOn a highly effective wave absorberrdquoCoastal Engineering Journal vol 15 no 4 pp 389ndash405 1991
[26] X P Yu ldquoDiffraction of water waves by porous breakwatersrdquoJournal of Waterway Port Coastal and Ocean Engineering vol121 no 6 pp 275ndash282 1995
[27] I J Losada M A Losada and A Baquerizo ldquoAn analyticalmethod to evaluate the efficiency of porous screens as wavedampersrdquo Applied Ocean Research vol 15 no 4 pp 207ndash2151993
[28] G Besio and M A Losada ldquoSediment transport patterns atTrafalgar offshore windfarmrdquo Ocean Engineering vol 35 no 7pp 653ndash665 2008
[29] H Song and L Tao ldquoShort-crestedwave interactionwith a con-centric porous cylindrical structurerdquo Applied Ocean Researchvol 29 no 4 pp 199ndash209 2007
[30] L Xiao Y Kou L Tao and L Yang ldquoComparative studyof hydrodynamic performances of breakwaters with double-layered perforated walls attached to ring-shaped very largefloating structuresrdquo Ocean Engineering vol 111 pp 279ndash2912016
[31] T L Yip and A T Chwang ldquoWater wave control by submergedpitching porous platerdquo Journal of Engineering Mechanics vol124 no 4 pp 428ndash434 1998
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
10 Mathematical Problems in Engineering
YHZheng et al(2004)
G=G=
G=G=
G=G=
G=G=infin
0
1
2
3
4
5
6m
2
as
963 100 87521 114kh
(a) Added mass
YH Zheng et al(2004)
G=G=infin
G=G=
G=G=
G=G=
963 100 87521 114kh
005
115
225
335
4
d
2
as
(b) Radiation coefficient
Figure 3 Variation of the dimensionless surge-surge hydrodynamic coefficients ((a) for the added mass and (b) for the radiation damping)versus 1198960ℎ under the conditions of different porous-effect parameters for 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16
YHZheng et al(2004)
G=G=
G=G=
G=G=
G=G=infin
963 100 87521 114kh
0
02
04
06
08
1
12
14
m
2
as
(a) Added mass
YHZheng et al(2004)
G=G=
G=G=
G=G=
G=G=infin
minus04minus02
002040608
11214
d
2
as
963 100 87521 114kh
(b) Radiation damping
Figure 4 Variation of the dimensionless heave-heave hydrodynamic coefficients ((a) for the added mass and (b) for the radiation damping)versus 1198960ℎ under the conditions of different porous-effect parameters for 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16
versus 1198960ℎ for the setting of 119904ℎ = 13 119889ℎ = 12 and 119886ℎ =16under the conditions of various porous-effect parametersWhen 11986601 = 11986602 = infin it represents the case withoutconsidering the effect of dual porous walls The presentanalytical solutions for this limiting case can be adopted tocompare to the limited numerical results available in thestudy of [7] The comparison plot is shown in Figure 4 wherethe nearly identical matches between the present analyticalsolutions and the numerical results of [7] can be observedThe results shown in Figure 4 suggest that the addition ofporous walls tends to enhance the added mass and dampingcoefficients In general with a decrease of porous effectparameter the addedmass and damping coefficients increaseFor the variation of addedmass coefficient with porouswalls
it can be seen that a peak value occurred at a long wave (small1198960ℎ value) condition However for the radiation dampingcoefficient it is shown to increase rapidly to its maximumvalue with an increase of 1198960ℎ then a gradual decrease trendis followed by a further increase of 1198960ℎ
The dimensionless heave-heave added mass (119898222120588119886119904)and radiation damping coefficient (119889222120588120596119886119904) versus 1198960ℎunder various 11986601 and 11986602 values for the case of 119904ℎ =13 119889ℎ = 12 119886ℎ = 16 are presented in Figure 4Under the limiting condition of 11986601 = 11986602 = infin (withoutporous walls) the numerical results produced by the study of[7] are also plotted for comparison Again good agreementsbetween the present analytical solutions and the publishednumerical results can be noticed The results in Figure 4(a)
Mathematical Problems in Engineering 11
YHZheng et al(2004)
963 100 87521 114kh
0
02
04
06
08
1
12
14m
2
as
G=G=infin
G=G=
G=G=
G=G=
(a) Added mass
YH Zhang et al(2004)
G=G=infin
G=G=
G=G=
G=G=
0010203040506070809
d
2
as
963 100 87521 114kh
(b) Radiation damping
Figure 5 Variation of the dimensionless pitch-pitch hydrodynamic coefficients ((a) for the added mass and (b) for the radiation damping)versus 1198960ℎ under the conditions of different porous-effect parameters for 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16 with the rotational center set at (00)
suggest that the variations of added mass for the shorterwavelength waves (large 1198960ℎ values) are less sensitive to thechange of porous parameters The presence of dual porouswalls increases the heave-heave added mass which tends toreach its maximum value at the long wave extreme Alsoa decrease in porous parameters results in an increase inheave induced added mass As shown in Figure 4(b) theheave-heave damping coefficients decrease rapidly when 1198960ℎincreases initially Then a gradual changing trend is followedby a further increase in 1198960ℎ Additionally the presence ofdual porous walls tends to reduce the values of heave-heavedamping coefficients
For the pitching induced added mass and dampingcoefficients the results are plotted against 1198960ℎ in Figure 5for various porous-effect parameters 11986601 and 11986602 and underthe settings of 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16 Whenthe rotational center is set at (00) as shown in Figure 5 thepresent analytical solutions match well with the publishednumerical results [7] for the limiting case of no porouswall (ie 11986601 = 11986602 = infin) It can be observed thatthe pitching oscillation induced added mass and dampingcoefficients for the floating breakwater system with dualporous side walls are greater than those without porous wallsWith decrease of the porous effect parameters the addedmass and damping coefficients increase For the added masscoefficients it is found that the variations trends are morecomplex for waves with relatively smaller 1198960ℎ under theconditions of different porous-effect parameters Generallyin cases of weaker porous-effect (for example 11986601 = 11986602 le 2as shown in Figure 5(a)) the value of added mass coefficienttends to approach 14 under the case considered For thevariations of damping coefficient as shown in Figure 5(b) thevarying trends are similar to those of damping coefficient insurge motion
The results presented in Figure 6 are the variations ofthe dimensionless surge-pitch added mass and damping
coefficients versus 1198960ℎ for varying 11986601 and 11986602 The case isfor 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16 with referenceto the rotational center of (00) Clearly the added massand damping coefficients are shown to increase with anincrease in the porous-effect parameters of both porouswalls A and B while the damping coefficients reach theirabsolute maximum values when 1198960ℎ is within the rangeof 1 and 2 The magnitude of surge-pitch hydrodynamiccoefficients especially the damping is greatly influenced bythe permeability of the dual porous walls compared to thesituation with surge or pitching motion only
The analytical solutions for the Response AmplitudeOperators (RAOs) of the floating breakwater system pro-posed in this study are illustrated in Figure 7 as a functionof dimensionless wavelength 1198960ℎ for various porous-effectparameters with the conditions of 119904ℎ = 03 119889ℎ = 05119886ℎ = 025 119903119866119886 = 09 and 119866119872119886 = 01 where therotational center is located at (00) It is found that the varyingtrends of surge RAOs are similar to those of pitching RAOsalthough the peak values of surge RAOs are smaller thanthose of pitching RAOs When the porous-effect parametersdecrease the main peak values of either surge or pitchingRAOs that occurred are shifted to the wave conditions withshorter wavelength Furthermore for the heave RAO itexperiences a typical damped response for the case of singledegree of freedom system and its maximum values of RAOwould move to the lower frequency wave region when theporous-effect parameters decrease
5 Summary and Conclusions
The hydrodynamic performance investigations includingthe hydrodynamic loads radiation induced added massand damping coefficients and dynamic responses relatedRAOs are critical to the study that is associated with theproblems of wave and structure interactions In this study the
12 Mathematical Problems in Engineering
G=G=infin
G=G=
G=G=
G=G=
963 100 87521 114kh
minus4
minus3
minus2
minus1
0
1
m
2
as
(a) Added mass
G=G=infin
G=G=
G=G=
G=G=
963 100 87521 114kh
minus3
minus25
minus2
minus15
minus1
minus05
0
d
2
as
(b) Radiation damping
Figure 6 Variation of the dimensionless surge-pitch hydrodynamic coefficients ((a) for the added mass and (b) for the radiation damping)versus 1198960ℎ under the conditions of different porous-effect parameters for 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16 with the rotational center set at(0 0)
G=G=
G=G=
G=G=
G=G=infin
4515 3525 3 605 2 5 551 4kh
0123456789
1011
|X|A
(a) Surge RAO
G=G=infin
G=G=
G=G=
G=G=
3 60 4 51 2kh
0
05
1
15
2
25
3
35|X
|A
(b) Heave RAO
G=G=
G=G=
G=G=
G=G=infin
05 3515 325 45 65 5541 20kh
0
5
10
15
|X|A
(c) Pitch RAO
Figure 7 Variation of the dimensionless RAO ((a) for surge RAO (b) for heave RAO and (c) for pitch RAO) versus 1198960ℎ under the conditionsof different porous-effect parameters for 119904ℎ = 03 119889ℎ = 05 119886ℎ = 025 119903119866119886 = 09 119866119872119886 = 01 with rotation center at (00)
Mathematical Problems in Engineering 13
hydrodynamic problems considering a floating breakwatersystem that consists of a rectangular body and two verticalattached porous walls are investigated analytically based onthe two-dimensional potential flow theory The expressionsof the diffraction and radiation based velocity potentials andthe associated unknown coefficients for each subdomain arederived by the MEEM approach Subsequently the excitinghydrodynamic forces added mass and radiation dampingcoefficients in terms of surge heave and pitching motionsand the equation ofmotion for the floating breakwater systemare formulated and calculated for various case scenariosby varying the wave conditions and porous effect param-eters Considering the limiting case without porous wallsgood agreements between the present analytical solutionsand published numerical results are achieved This suggeststhat the present analytical model is demonstrated to be areliable engineering tool to provide reasonable estimationsfor the hydrodynamic performances of a floating breakwatersystem with dual side porous walls It is worthy to noticethat the wave loads on the floating structures especiallythe moments when referenced to the rotational center ofbottom center would be reduced with an increase in theporous-effect parameters of the porous walls Additionallythe existence of the dual porous walls could move the peaksof surge and pitching RAOs to the waves of higher frequencywhen the porous effect parameters decrease Overall it canbe concluded that a properly adopted porous-wall systemwhen attached to a floating body can effectively improveits hydrodynamic performance as a breakwater The analyt-ical approach developed in this study can also be furtherextended to assist the theoretical analysis for other similar butimproved floating breakwater systems
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was supported by ldquoFundamental Research Fundsfor the Central Universitiesrdquo (Grand No 017180224)
References
[1] I-H Cho ldquoTransmission coefficients of a floating rectangularbreakwater with porous side platesrdquo International Journal ofNaval Architecture and Ocean Engineering vol 8 no 1 pp 53ndash65 2016
[2] H Hu K-H Wang and A N Williams ldquoWave motion over abreakwater system of a horizontal plate and a vertical porouswallrdquo Ocean Engineering vol 29 no 4 pp 373ndash386 2001
[3] C C Mei and J L Black ldquoScattering of surface waves byrectangular obstacles in water of finite depthrdquo Journal of FluidMechanics vol 38 no 3 pp 499ndash511 1969
[4] J L Black C C Mei and M C G Bray ldquoRadiation andscattering of water waves by rigid bodiesrdquo Journal of FluidMechanics vol 46 no 1 pp 151ndash164 1971
[5] N Drimer Y Agnon and M Stiassnie ldquoA simplified analyticalmodel for a floating breakwater in water of finite depthrdquoAppliedOcean Research vol 14 no 1 pp 33ndash41 1992
[6] J-F Lee ldquoOn the heave radiation of a rectangular structurerdquoOcean Engineering vol 22 no 1 pp 19ndash34 1995
[7] Y H Zheng Y G You and Y M Shen ldquoOn the radiationand diffraction of water waves by a rectangular buoyrdquo OceanEngineering vol 31 no 8-9 pp 1063ndash1082 2004
[8] E D Christensen H B Bingham A P Skou Friis A K Larsenand K L Jensen ldquoAn experimental and numerical study offloating breakwatersrdquo Coastal Engineering Journal vol 137 pp43ndash58 2018
[9] E Koutandos P Prinos andXGironella ldquoFloating breakwatersunder regular and irregular wave forcing Reflection and trans-mission characteristicsrdquo Journal of Hydraulic Research vol 43no 2 pp 174ndash188 2005
[10] G H Dong Y N Zheng Y C Li B Teng C T Guan and D FLin ldquoExperiments on wave transmission coefficients of floatingbreakwatersrdquo Ocean Engineering vol 35 no 8-9 pp 931ndash9382008
[11] A T Chwang ldquoA porous-wavemaker theoryrdquo Journal of FluidMechanics vol 132 pp 395ndash406 1983
[12] Y C Li Y Liu and B Teng ldquoPorous effect parameter of thinpermeable platesrdquo Coastal Engineering Journal vol 48 no 4pp 309ndash336 2006
[13] S R Manam and M Sivanesan ldquoScattering of water wavesby vertical porous barriers an analytical approachrdquo WaveMotion An International Journal Reporting Research on WavePhenomena vol 67 pp 89ndash101 2016
[14] A T Chwang and Z Dong ldquoWave-trapping due to a porousplaterdquo in Proceedings of 15 ONR symposium on Naval Hydrody-namics pp 407ndash414 1985
[15] S Zhu and A T Chwang ldquoAnalytical study of porous waveabsorberrdquo Journal of Engineering Mechanics vol 127 no 4 pp326ndash332 2001
[16] H Behera and T Sahoo ldquoHydroelastic analysis of gravity waveinteraction with submerged horizontal flexible porous platerdquoJournal of Fluids and Structures vol 54 pp 643ndash660 2015
[17] T L Yip and A T Chwang ldquoPerforated wall breakwater withinternal horizontal platerdquo Journal of Engineering Mechanics vol126 no 5 pp 533ndash538 2000
[18] Y Liu and H-J Li ldquoWave scattering by dual submergedhorizontal porous plates Further resultsrdquo Ocean Engineeringvol 81 no 5 pp 158ndash163 2014
[19] I H Cho and M H Kim ldquoWave absorbing system usinginclined perforated platesrdquo Journal of Fluid Mechanics vol 608pp 1ndash20 2008
[20] K-H Wang and X Ren ldquoWater waves on flexible and porousbreakwatersrdquo Journal of Engineering Mechanics vol 119 no 5pp 1025ndash1047 1993
[21] K H Wang and X Ren ldquoWave interaction with a concentricporous cylinder systemrdquo Ocean Engineering vol 21 no 4 pp343ndash360 1994
[22] F Zhao T Zhang R Wan L Huang X Wang and WBao ldquoHydrodynamic loads acting on a circular porous platehorizontally submerged in wavesrdquo Ocean Engineering vol 136pp 168ndash177 2017
14 Mathematical Problems in Engineering
[23] A Mondal and R Gayen ldquoWave interaction with dual circularporous platesrdquo Journal of Marine Science and Application vol14 no 4 pp 366ndash375 2015
[24] W L Qiao K-H Wang and Y Q Sun ldquoScattering of waterwaves by a floating body with two vertically attached porouswallsrdquo Journal of Engineering Mechanics vol 144 no 2 ArticleID 04017162 2018
[25] S W Twu and D T Lin ldquoOn a highly effective wave absorberrdquoCoastal Engineering Journal vol 15 no 4 pp 389ndash405 1991
[26] X P Yu ldquoDiffraction of water waves by porous breakwatersrdquoJournal of Waterway Port Coastal and Ocean Engineering vol121 no 6 pp 275ndash282 1995
[27] I J Losada M A Losada and A Baquerizo ldquoAn analyticalmethod to evaluate the efficiency of porous screens as wavedampersrdquo Applied Ocean Research vol 15 no 4 pp 207ndash2151993
[28] G Besio and M A Losada ldquoSediment transport patterns atTrafalgar offshore windfarmrdquo Ocean Engineering vol 35 no 7pp 653ndash665 2008
[29] H Song and L Tao ldquoShort-crestedwave interactionwith a con-centric porous cylindrical structurerdquo Applied Ocean Researchvol 29 no 4 pp 199ndash209 2007
[30] L Xiao Y Kou L Tao and L Yang ldquoComparative studyof hydrodynamic performances of breakwaters with double-layered perforated walls attached to ring-shaped very largefloating structuresrdquo Ocean Engineering vol 111 pp 279ndash2912016
[31] T L Yip and A T Chwang ldquoWater wave control by submergedpitching porous platerdquo Journal of Engineering Mechanics vol124 no 4 pp 428ndash434 1998
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Mathematical Problems in Engineering 11
YHZheng et al(2004)
963 100 87521 114kh
0
02
04
06
08
1
12
14m
2
as
G=G=infin
G=G=
G=G=
G=G=
(a) Added mass
YH Zhang et al(2004)
G=G=infin
G=G=
G=G=
G=G=
0010203040506070809
d
2
as
963 100 87521 114kh
(b) Radiation damping
Figure 5 Variation of the dimensionless pitch-pitch hydrodynamic coefficients ((a) for the added mass and (b) for the radiation damping)versus 1198960ℎ under the conditions of different porous-effect parameters for 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16 with the rotational center set at (00)
suggest that the variations of added mass for the shorterwavelength waves (large 1198960ℎ values) are less sensitive to thechange of porous parameters The presence of dual porouswalls increases the heave-heave added mass which tends toreach its maximum value at the long wave extreme Alsoa decrease in porous parameters results in an increase inheave induced added mass As shown in Figure 4(b) theheave-heave damping coefficients decrease rapidly when 1198960ℎincreases initially Then a gradual changing trend is followedby a further increase in 1198960ℎ Additionally the presence ofdual porous walls tends to reduce the values of heave-heavedamping coefficients
For the pitching induced added mass and dampingcoefficients the results are plotted against 1198960ℎ in Figure 5for various porous-effect parameters 11986601 and 11986602 and underthe settings of 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16 Whenthe rotational center is set at (00) as shown in Figure 5 thepresent analytical solutions match well with the publishednumerical results [7] for the limiting case of no porouswall (ie 11986601 = 11986602 = infin) It can be observed thatthe pitching oscillation induced added mass and dampingcoefficients for the floating breakwater system with dualporous side walls are greater than those without porous wallsWith decrease of the porous effect parameters the addedmass and damping coefficients increase For the added masscoefficients it is found that the variations trends are morecomplex for waves with relatively smaller 1198960ℎ under theconditions of different porous-effect parameters Generallyin cases of weaker porous-effect (for example 11986601 = 11986602 le 2as shown in Figure 5(a)) the value of added mass coefficienttends to approach 14 under the case considered For thevariations of damping coefficient as shown in Figure 5(b) thevarying trends are similar to those of damping coefficient insurge motion
The results presented in Figure 6 are the variations ofthe dimensionless surge-pitch added mass and damping
coefficients versus 1198960ℎ for varying 11986601 and 11986602 The case isfor 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16 with referenceto the rotational center of (00) Clearly the added massand damping coefficients are shown to increase with anincrease in the porous-effect parameters of both porouswalls A and B while the damping coefficients reach theirabsolute maximum values when 1198960ℎ is within the rangeof 1 and 2 The magnitude of surge-pitch hydrodynamiccoefficients especially the damping is greatly influenced bythe permeability of the dual porous walls compared to thesituation with surge or pitching motion only
The analytical solutions for the Response AmplitudeOperators (RAOs) of the floating breakwater system pro-posed in this study are illustrated in Figure 7 as a functionof dimensionless wavelength 1198960ℎ for various porous-effectparameters with the conditions of 119904ℎ = 03 119889ℎ = 05119886ℎ = 025 119903119866119886 = 09 and 119866119872119886 = 01 where therotational center is located at (00) It is found that the varyingtrends of surge RAOs are similar to those of pitching RAOsalthough the peak values of surge RAOs are smaller thanthose of pitching RAOs When the porous-effect parametersdecrease the main peak values of either surge or pitchingRAOs that occurred are shifted to the wave conditions withshorter wavelength Furthermore for the heave RAO itexperiences a typical damped response for the case of singledegree of freedom system and its maximum values of RAOwould move to the lower frequency wave region when theporous-effect parameters decrease
5 Summary and Conclusions
The hydrodynamic performance investigations includingthe hydrodynamic loads radiation induced added massand damping coefficients and dynamic responses relatedRAOs are critical to the study that is associated with theproblems of wave and structure interactions In this study the
12 Mathematical Problems in Engineering
G=G=infin
G=G=
G=G=
G=G=
963 100 87521 114kh
minus4
minus3
minus2
minus1
0
1
m
2
as
(a) Added mass
G=G=infin
G=G=
G=G=
G=G=
963 100 87521 114kh
minus3
minus25
minus2
minus15
minus1
minus05
0
d
2
as
(b) Radiation damping
Figure 6 Variation of the dimensionless surge-pitch hydrodynamic coefficients ((a) for the added mass and (b) for the radiation damping)versus 1198960ℎ under the conditions of different porous-effect parameters for 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16 with the rotational center set at(0 0)
G=G=
G=G=
G=G=
G=G=infin
4515 3525 3 605 2 5 551 4kh
0123456789
1011
|X|A
(a) Surge RAO
G=G=infin
G=G=
G=G=
G=G=
3 60 4 51 2kh
0
05
1
15
2
25
3
35|X
|A
(b) Heave RAO
G=G=
G=G=
G=G=
G=G=infin
05 3515 325 45 65 5541 20kh
0
5
10
15
|X|A
(c) Pitch RAO
Figure 7 Variation of the dimensionless RAO ((a) for surge RAO (b) for heave RAO and (c) for pitch RAO) versus 1198960ℎ under the conditionsof different porous-effect parameters for 119904ℎ = 03 119889ℎ = 05 119886ℎ = 025 119903119866119886 = 09 119866119872119886 = 01 with rotation center at (00)
Mathematical Problems in Engineering 13
hydrodynamic problems considering a floating breakwatersystem that consists of a rectangular body and two verticalattached porous walls are investigated analytically based onthe two-dimensional potential flow theory The expressionsof the diffraction and radiation based velocity potentials andthe associated unknown coefficients for each subdomain arederived by the MEEM approach Subsequently the excitinghydrodynamic forces added mass and radiation dampingcoefficients in terms of surge heave and pitching motionsand the equation ofmotion for the floating breakwater systemare formulated and calculated for various case scenariosby varying the wave conditions and porous effect param-eters Considering the limiting case without porous wallsgood agreements between the present analytical solutionsand published numerical results are achieved This suggeststhat the present analytical model is demonstrated to be areliable engineering tool to provide reasonable estimationsfor the hydrodynamic performances of a floating breakwatersystem with dual side porous walls It is worthy to noticethat the wave loads on the floating structures especiallythe moments when referenced to the rotational center ofbottom center would be reduced with an increase in theporous-effect parameters of the porous walls Additionallythe existence of the dual porous walls could move the peaksof surge and pitching RAOs to the waves of higher frequencywhen the porous effect parameters decrease Overall it canbe concluded that a properly adopted porous-wall systemwhen attached to a floating body can effectively improveits hydrodynamic performance as a breakwater The analyt-ical approach developed in this study can also be furtherextended to assist the theoretical analysis for other similar butimproved floating breakwater systems
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was supported by ldquoFundamental Research Fundsfor the Central Universitiesrdquo (Grand No 017180224)
References
[1] I-H Cho ldquoTransmission coefficients of a floating rectangularbreakwater with porous side platesrdquo International Journal ofNaval Architecture and Ocean Engineering vol 8 no 1 pp 53ndash65 2016
[2] H Hu K-H Wang and A N Williams ldquoWave motion over abreakwater system of a horizontal plate and a vertical porouswallrdquo Ocean Engineering vol 29 no 4 pp 373ndash386 2001
[3] C C Mei and J L Black ldquoScattering of surface waves byrectangular obstacles in water of finite depthrdquo Journal of FluidMechanics vol 38 no 3 pp 499ndash511 1969
[4] J L Black C C Mei and M C G Bray ldquoRadiation andscattering of water waves by rigid bodiesrdquo Journal of FluidMechanics vol 46 no 1 pp 151ndash164 1971
[5] N Drimer Y Agnon and M Stiassnie ldquoA simplified analyticalmodel for a floating breakwater in water of finite depthrdquoAppliedOcean Research vol 14 no 1 pp 33ndash41 1992
[6] J-F Lee ldquoOn the heave radiation of a rectangular structurerdquoOcean Engineering vol 22 no 1 pp 19ndash34 1995
[7] Y H Zheng Y G You and Y M Shen ldquoOn the radiationand diffraction of water waves by a rectangular buoyrdquo OceanEngineering vol 31 no 8-9 pp 1063ndash1082 2004
[8] E D Christensen H B Bingham A P Skou Friis A K Larsenand K L Jensen ldquoAn experimental and numerical study offloating breakwatersrdquo Coastal Engineering Journal vol 137 pp43ndash58 2018
[9] E Koutandos P Prinos andXGironella ldquoFloating breakwatersunder regular and irregular wave forcing Reflection and trans-mission characteristicsrdquo Journal of Hydraulic Research vol 43no 2 pp 174ndash188 2005
[10] G H Dong Y N Zheng Y C Li B Teng C T Guan and D FLin ldquoExperiments on wave transmission coefficients of floatingbreakwatersrdquo Ocean Engineering vol 35 no 8-9 pp 931ndash9382008
[11] A T Chwang ldquoA porous-wavemaker theoryrdquo Journal of FluidMechanics vol 132 pp 395ndash406 1983
[12] Y C Li Y Liu and B Teng ldquoPorous effect parameter of thinpermeable platesrdquo Coastal Engineering Journal vol 48 no 4pp 309ndash336 2006
[13] S R Manam and M Sivanesan ldquoScattering of water wavesby vertical porous barriers an analytical approachrdquo WaveMotion An International Journal Reporting Research on WavePhenomena vol 67 pp 89ndash101 2016
[14] A T Chwang and Z Dong ldquoWave-trapping due to a porousplaterdquo in Proceedings of 15 ONR symposium on Naval Hydrody-namics pp 407ndash414 1985
[15] S Zhu and A T Chwang ldquoAnalytical study of porous waveabsorberrdquo Journal of Engineering Mechanics vol 127 no 4 pp326ndash332 2001
[16] H Behera and T Sahoo ldquoHydroelastic analysis of gravity waveinteraction with submerged horizontal flexible porous platerdquoJournal of Fluids and Structures vol 54 pp 643ndash660 2015
[17] T L Yip and A T Chwang ldquoPerforated wall breakwater withinternal horizontal platerdquo Journal of Engineering Mechanics vol126 no 5 pp 533ndash538 2000
[18] Y Liu and H-J Li ldquoWave scattering by dual submergedhorizontal porous plates Further resultsrdquo Ocean Engineeringvol 81 no 5 pp 158ndash163 2014
[19] I H Cho and M H Kim ldquoWave absorbing system usinginclined perforated platesrdquo Journal of Fluid Mechanics vol 608pp 1ndash20 2008
[20] K-H Wang and X Ren ldquoWater waves on flexible and porousbreakwatersrdquo Journal of Engineering Mechanics vol 119 no 5pp 1025ndash1047 1993
[21] K H Wang and X Ren ldquoWave interaction with a concentricporous cylinder systemrdquo Ocean Engineering vol 21 no 4 pp343ndash360 1994
[22] F Zhao T Zhang R Wan L Huang X Wang and WBao ldquoHydrodynamic loads acting on a circular porous platehorizontally submerged in wavesrdquo Ocean Engineering vol 136pp 168ndash177 2017
14 Mathematical Problems in Engineering
[23] A Mondal and R Gayen ldquoWave interaction with dual circularporous platesrdquo Journal of Marine Science and Application vol14 no 4 pp 366ndash375 2015
[24] W L Qiao K-H Wang and Y Q Sun ldquoScattering of waterwaves by a floating body with two vertically attached porouswallsrdquo Journal of Engineering Mechanics vol 144 no 2 ArticleID 04017162 2018
[25] S W Twu and D T Lin ldquoOn a highly effective wave absorberrdquoCoastal Engineering Journal vol 15 no 4 pp 389ndash405 1991
[26] X P Yu ldquoDiffraction of water waves by porous breakwatersrdquoJournal of Waterway Port Coastal and Ocean Engineering vol121 no 6 pp 275ndash282 1995
[27] I J Losada M A Losada and A Baquerizo ldquoAn analyticalmethod to evaluate the efficiency of porous screens as wavedampersrdquo Applied Ocean Research vol 15 no 4 pp 207ndash2151993
[28] G Besio and M A Losada ldquoSediment transport patterns atTrafalgar offshore windfarmrdquo Ocean Engineering vol 35 no 7pp 653ndash665 2008
[29] H Song and L Tao ldquoShort-crestedwave interactionwith a con-centric porous cylindrical structurerdquo Applied Ocean Researchvol 29 no 4 pp 199ndash209 2007
[30] L Xiao Y Kou L Tao and L Yang ldquoComparative studyof hydrodynamic performances of breakwaters with double-layered perforated walls attached to ring-shaped very largefloating structuresrdquo Ocean Engineering vol 111 pp 279ndash2912016
[31] T L Yip and A T Chwang ldquoWater wave control by submergedpitching porous platerdquo Journal of Engineering Mechanics vol124 no 4 pp 428ndash434 1998
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
12 Mathematical Problems in Engineering
G=G=infin
G=G=
G=G=
G=G=
963 100 87521 114kh
minus4
minus3
minus2
minus1
0
1
m
2
as
(a) Added mass
G=G=infin
G=G=
G=G=
G=G=
963 100 87521 114kh
minus3
minus25
minus2
minus15
minus1
minus05
0
d
2
as
(b) Radiation damping
Figure 6 Variation of the dimensionless surge-pitch hydrodynamic coefficients ((a) for the added mass and (b) for the radiation damping)versus 1198960ℎ under the conditions of different porous-effect parameters for 119904ℎ = 13 119889ℎ = 12 119886ℎ = 16 with the rotational center set at(0 0)
G=G=
G=G=
G=G=
G=G=infin
4515 3525 3 605 2 5 551 4kh
0123456789
1011
|X|A
(a) Surge RAO
G=G=infin
G=G=
G=G=
G=G=
3 60 4 51 2kh
0
05
1
15
2
25
3
35|X
|A
(b) Heave RAO
G=G=
G=G=
G=G=
G=G=infin
05 3515 325 45 65 5541 20kh
0
5
10
15
|X|A
(c) Pitch RAO
Figure 7 Variation of the dimensionless RAO ((a) for surge RAO (b) for heave RAO and (c) for pitch RAO) versus 1198960ℎ under the conditionsof different porous-effect parameters for 119904ℎ = 03 119889ℎ = 05 119886ℎ = 025 119903119866119886 = 09 119866119872119886 = 01 with rotation center at (00)
Mathematical Problems in Engineering 13
hydrodynamic problems considering a floating breakwatersystem that consists of a rectangular body and two verticalattached porous walls are investigated analytically based onthe two-dimensional potential flow theory The expressionsof the diffraction and radiation based velocity potentials andthe associated unknown coefficients for each subdomain arederived by the MEEM approach Subsequently the excitinghydrodynamic forces added mass and radiation dampingcoefficients in terms of surge heave and pitching motionsand the equation ofmotion for the floating breakwater systemare formulated and calculated for various case scenariosby varying the wave conditions and porous effect param-eters Considering the limiting case without porous wallsgood agreements between the present analytical solutionsand published numerical results are achieved This suggeststhat the present analytical model is demonstrated to be areliable engineering tool to provide reasonable estimationsfor the hydrodynamic performances of a floating breakwatersystem with dual side porous walls It is worthy to noticethat the wave loads on the floating structures especiallythe moments when referenced to the rotational center ofbottom center would be reduced with an increase in theporous-effect parameters of the porous walls Additionallythe existence of the dual porous walls could move the peaksof surge and pitching RAOs to the waves of higher frequencywhen the porous effect parameters decrease Overall it canbe concluded that a properly adopted porous-wall systemwhen attached to a floating body can effectively improveits hydrodynamic performance as a breakwater The analyt-ical approach developed in this study can also be furtherextended to assist the theoretical analysis for other similar butimproved floating breakwater systems
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was supported by ldquoFundamental Research Fundsfor the Central Universitiesrdquo (Grand No 017180224)
References
[1] I-H Cho ldquoTransmission coefficients of a floating rectangularbreakwater with porous side platesrdquo International Journal ofNaval Architecture and Ocean Engineering vol 8 no 1 pp 53ndash65 2016
[2] H Hu K-H Wang and A N Williams ldquoWave motion over abreakwater system of a horizontal plate and a vertical porouswallrdquo Ocean Engineering vol 29 no 4 pp 373ndash386 2001
[3] C C Mei and J L Black ldquoScattering of surface waves byrectangular obstacles in water of finite depthrdquo Journal of FluidMechanics vol 38 no 3 pp 499ndash511 1969
[4] J L Black C C Mei and M C G Bray ldquoRadiation andscattering of water waves by rigid bodiesrdquo Journal of FluidMechanics vol 46 no 1 pp 151ndash164 1971
[5] N Drimer Y Agnon and M Stiassnie ldquoA simplified analyticalmodel for a floating breakwater in water of finite depthrdquoAppliedOcean Research vol 14 no 1 pp 33ndash41 1992
[6] J-F Lee ldquoOn the heave radiation of a rectangular structurerdquoOcean Engineering vol 22 no 1 pp 19ndash34 1995
[7] Y H Zheng Y G You and Y M Shen ldquoOn the radiationand diffraction of water waves by a rectangular buoyrdquo OceanEngineering vol 31 no 8-9 pp 1063ndash1082 2004
[8] E D Christensen H B Bingham A P Skou Friis A K Larsenand K L Jensen ldquoAn experimental and numerical study offloating breakwatersrdquo Coastal Engineering Journal vol 137 pp43ndash58 2018
[9] E Koutandos P Prinos andXGironella ldquoFloating breakwatersunder regular and irregular wave forcing Reflection and trans-mission characteristicsrdquo Journal of Hydraulic Research vol 43no 2 pp 174ndash188 2005
[10] G H Dong Y N Zheng Y C Li B Teng C T Guan and D FLin ldquoExperiments on wave transmission coefficients of floatingbreakwatersrdquo Ocean Engineering vol 35 no 8-9 pp 931ndash9382008
[11] A T Chwang ldquoA porous-wavemaker theoryrdquo Journal of FluidMechanics vol 132 pp 395ndash406 1983
[12] Y C Li Y Liu and B Teng ldquoPorous effect parameter of thinpermeable platesrdquo Coastal Engineering Journal vol 48 no 4pp 309ndash336 2006
[13] S R Manam and M Sivanesan ldquoScattering of water wavesby vertical porous barriers an analytical approachrdquo WaveMotion An International Journal Reporting Research on WavePhenomena vol 67 pp 89ndash101 2016
[14] A T Chwang and Z Dong ldquoWave-trapping due to a porousplaterdquo in Proceedings of 15 ONR symposium on Naval Hydrody-namics pp 407ndash414 1985
[15] S Zhu and A T Chwang ldquoAnalytical study of porous waveabsorberrdquo Journal of Engineering Mechanics vol 127 no 4 pp326ndash332 2001
[16] H Behera and T Sahoo ldquoHydroelastic analysis of gravity waveinteraction with submerged horizontal flexible porous platerdquoJournal of Fluids and Structures vol 54 pp 643ndash660 2015
[17] T L Yip and A T Chwang ldquoPerforated wall breakwater withinternal horizontal platerdquo Journal of Engineering Mechanics vol126 no 5 pp 533ndash538 2000
[18] Y Liu and H-J Li ldquoWave scattering by dual submergedhorizontal porous plates Further resultsrdquo Ocean Engineeringvol 81 no 5 pp 158ndash163 2014
[19] I H Cho and M H Kim ldquoWave absorbing system usinginclined perforated platesrdquo Journal of Fluid Mechanics vol 608pp 1ndash20 2008
[20] K-H Wang and X Ren ldquoWater waves on flexible and porousbreakwatersrdquo Journal of Engineering Mechanics vol 119 no 5pp 1025ndash1047 1993
[21] K H Wang and X Ren ldquoWave interaction with a concentricporous cylinder systemrdquo Ocean Engineering vol 21 no 4 pp343ndash360 1994
[22] F Zhao T Zhang R Wan L Huang X Wang and WBao ldquoHydrodynamic loads acting on a circular porous platehorizontally submerged in wavesrdquo Ocean Engineering vol 136pp 168ndash177 2017
14 Mathematical Problems in Engineering
[23] A Mondal and R Gayen ldquoWave interaction with dual circularporous platesrdquo Journal of Marine Science and Application vol14 no 4 pp 366ndash375 2015
[24] W L Qiao K-H Wang and Y Q Sun ldquoScattering of waterwaves by a floating body with two vertically attached porouswallsrdquo Journal of Engineering Mechanics vol 144 no 2 ArticleID 04017162 2018
[25] S W Twu and D T Lin ldquoOn a highly effective wave absorberrdquoCoastal Engineering Journal vol 15 no 4 pp 389ndash405 1991
[26] X P Yu ldquoDiffraction of water waves by porous breakwatersrdquoJournal of Waterway Port Coastal and Ocean Engineering vol121 no 6 pp 275ndash282 1995
[27] I J Losada M A Losada and A Baquerizo ldquoAn analyticalmethod to evaluate the efficiency of porous screens as wavedampersrdquo Applied Ocean Research vol 15 no 4 pp 207ndash2151993
[28] G Besio and M A Losada ldquoSediment transport patterns atTrafalgar offshore windfarmrdquo Ocean Engineering vol 35 no 7pp 653ndash665 2008
[29] H Song and L Tao ldquoShort-crestedwave interactionwith a con-centric porous cylindrical structurerdquo Applied Ocean Researchvol 29 no 4 pp 199ndash209 2007
[30] L Xiao Y Kou L Tao and L Yang ldquoComparative studyof hydrodynamic performances of breakwaters with double-layered perforated walls attached to ring-shaped very largefloating structuresrdquo Ocean Engineering vol 111 pp 279ndash2912016
[31] T L Yip and A T Chwang ldquoWater wave control by submergedpitching porous platerdquo Journal of Engineering Mechanics vol124 no 4 pp 428ndash434 1998
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Mathematical Problems in Engineering 13
hydrodynamic problems considering a floating breakwatersystem that consists of a rectangular body and two verticalattached porous walls are investigated analytically based onthe two-dimensional potential flow theory The expressionsof the diffraction and radiation based velocity potentials andthe associated unknown coefficients for each subdomain arederived by the MEEM approach Subsequently the excitinghydrodynamic forces added mass and radiation dampingcoefficients in terms of surge heave and pitching motionsand the equation ofmotion for the floating breakwater systemare formulated and calculated for various case scenariosby varying the wave conditions and porous effect param-eters Considering the limiting case without porous wallsgood agreements between the present analytical solutionsand published numerical results are achieved This suggeststhat the present analytical model is demonstrated to be areliable engineering tool to provide reasonable estimationsfor the hydrodynamic performances of a floating breakwatersystem with dual side porous walls It is worthy to noticethat the wave loads on the floating structures especiallythe moments when referenced to the rotational center ofbottom center would be reduced with an increase in theporous-effect parameters of the porous walls Additionallythe existence of the dual porous walls could move the peaksof surge and pitching RAOs to the waves of higher frequencywhen the porous effect parameters decrease Overall it canbe concluded that a properly adopted porous-wall systemwhen attached to a floating body can effectively improveits hydrodynamic performance as a breakwater The analyt-ical approach developed in this study can also be furtherextended to assist the theoretical analysis for other similar butimproved floating breakwater systems
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was supported by ldquoFundamental Research Fundsfor the Central Universitiesrdquo (Grand No 017180224)
References
[1] I-H Cho ldquoTransmission coefficients of a floating rectangularbreakwater with porous side platesrdquo International Journal ofNaval Architecture and Ocean Engineering vol 8 no 1 pp 53ndash65 2016
[2] H Hu K-H Wang and A N Williams ldquoWave motion over abreakwater system of a horizontal plate and a vertical porouswallrdquo Ocean Engineering vol 29 no 4 pp 373ndash386 2001
[3] C C Mei and J L Black ldquoScattering of surface waves byrectangular obstacles in water of finite depthrdquo Journal of FluidMechanics vol 38 no 3 pp 499ndash511 1969
[4] J L Black C C Mei and M C G Bray ldquoRadiation andscattering of water waves by rigid bodiesrdquo Journal of FluidMechanics vol 46 no 1 pp 151ndash164 1971
[5] N Drimer Y Agnon and M Stiassnie ldquoA simplified analyticalmodel for a floating breakwater in water of finite depthrdquoAppliedOcean Research vol 14 no 1 pp 33ndash41 1992
[6] J-F Lee ldquoOn the heave radiation of a rectangular structurerdquoOcean Engineering vol 22 no 1 pp 19ndash34 1995
[7] Y H Zheng Y G You and Y M Shen ldquoOn the radiationand diffraction of water waves by a rectangular buoyrdquo OceanEngineering vol 31 no 8-9 pp 1063ndash1082 2004
[8] E D Christensen H B Bingham A P Skou Friis A K Larsenand K L Jensen ldquoAn experimental and numerical study offloating breakwatersrdquo Coastal Engineering Journal vol 137 pp43ndash58 2018
[9] E Koutandos P Prinos andXGironella ldquoFloating breakwatersunder regular and irregular wave forcing Reflection and trans-mission characteristicsrdquo Journal of Hydraulic Research vol 43no 2 pp 174ndash188 2005
[10] G H Dong Y N Zheng Y C Li B Teng C T Guan and D FLin ldquoExperiments on wave transmission coefficients of floatingbreakwatersrdquo Ocean Engineering vol 35 no 8-9 pp 931ndash9382008
[11] A T Chwang ldquoA porous-wavemaker theoryrdquo Journal of FluidMechanics vol 132 pp 395ndash406 1983
[12] Y C Li Y Liu and B Teng ldquoPorous effect parameter of thinpermeable platesrdquo Coastal Engineering Journal vol 48 no 4pp 309ndash336 2006
[13] S R Manam and M Sivanesan ldquoScattering of water wavesby vertical porous barriers an analytical approachrdquo WaveMotion An International Journal Reporting Research on WavePhenomena vol 67 pp 89ndash101 2016
[14] A T Chwang and Z Dong ldquoWave-trapping due to a porousplaterdquo in Proceedings of 15 ONR symposium on Naval Hydrody-namics pp 407ndash414 1985
[15] S Zhu and A T Chwang ldquoAnalytical study of porous waveabsorberrdquo Journal of Engineering Mechanics vol 127 no 4 pp326ndash332 2001
[16] H Behera and T Sahoo ldquoHydroelastic analysis of gravity waveinteraction with submerged horizontal flexible porous platerdquoJournal of Fluids and Structures vol 54 pp 643ndash660 2015
[17] T L Yip and A T Chwang ldquoPerforated wall breakwater withinternal horizontal platerdquo Journal of Engineering Mechanics vol126 no 5 pp 533ndash538 2000
[18] Y Liu and H-J Li ldquoWave scattering by dual submergedhorizontal porous plates Further resultsrdquo Ocean Engineeringvol 81 no 5 pp 158ndash163 2014
[19] I H Cho and M H Kim ldquoWave absorbing system usinginclined perforated platesrdquo Journal of Fluid Mechanics vol 608pp 1ndash20 2008
[20] K-H Wang and X Ren ldquoWater waves on flexible and porousbreakwatersrdquo Journal of Engineering Mechanics vol 119 no 5pp 1025ndash1047 1993
[21] K H Wang and X Ren ldquoWave interaction with a concentricporous cylinder systemrdquo Ocean Engineering vol 21 no 4 pp343ndash360 1994
[22] F Zhao T Zhang R Wan L Huang X Wang and WBao ldquoHydrodynamic loads acting on a circular porous platehorizontally submerged in wavesrdquo Ocean Engineering vol 136pp 168ndash177 2017
14 Mathematical Problems in Engineering
[23] A Mondal and R Gayen ldquoWave interaction with dual circularporous platesrdquo Journal of Marine Science and Application vol14 no 4 pp 366ndash375 2015
[24] W L Qiao K-H Wang and Y Q Sun ldquoScattering of waterwaves by a floating body with two vertically attached porouswallsrdquo Journal of Engineering Mechanics vol 144 no 2 ArticleID 04017162 2018
[25] S W Twu and D T Lin ldquoOn a highly effective wave absorberrdquoCoastal Engineering Journal vol 15 no 4 pp 389ndash405 1991
[26] X P Yu ldquoDiffraction of water waves by porous breakwatersrdquoJournal of Waterway Port Coastal and Ocean Engineering vol121 no 6 pp 275ndash282 1995
[27] I J Losada M A Losada and A Baquerizo ldquoAn analyticalmethod to evaluate the efficiency of porous screens as wavedampersrdquo Applied Ocean Research vol 15 no 4 pp 207ndash2151993
[28] G Besio and M A Losada ldquoSediment transport patterns atTrafalgar offshore windfarmrdquo Ocean Engineering vol 35 no 7pp 653ndash665 2008
[29] H Song and L Tao ldquoShort-crestedwave interactionwith a con-centric porous cylindrical structurerdquo Applied Ocean Researchvol 29 no 4 pp 199ndash209 2007
[30] L Xiao Y Kou L Tao and L Yang ldquoComparative studyof hydrodynamic performances of breakwaters with double-layered perforated walls attached to ring-shaped very largefloating structuresrdquo Ocean Engineering vol 111 pp 279ndash2912016
[31] T L Yip and A T Chwang ldquoWater wave control by submergedpitching porous platerdquo Journal of Engineering Mechanics vol124 no 4 pp 428ndash434 1998
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
14 Mathematical Problems in Engineering
[23] A Mondal and R Gayen ldquoWave interaction with dual circularporous platesrdquo Journal of Marine Science and Application vol14 no 4 pp 366ndash375 2015
[24] W L Qiao K-H Wang and Y Q Sun ldquoScattering of waterwaves by a floating body with two vertically attached porouswallsrdquo Journal of Engineering Mechanics vol 144 no 2 ArticleID 04017162 2018
[25] S W Twu and D T Lin ldquoOn a highly effective wave absorberrdquoCoastal Engineering Journal vol 15 no 4 pp 389ndash405 1991
[26] X P Yu ldquoDiffraction of water waves by porous breakwatersrdquoJournal of Waterway Port Coastal and Ocean Engineering vol121 no 6 pp 275ndash282 1995
[27] I J Losada M A Losada and A Baquerizo ldquoAn analyticalmethod to evaluate the efficiency of porous screens as wavedampersrdquo Applied Ocean Research vol 15 no 4 pp 207ndash2151993
[28] G Besio and M A Losada ldquoSediment transport patterns atTrafalgar offshore windfarmrdquo Ocean Engineering vol 35 no 7pp 653ndash665 2008
[29] H Song and L Tao ldquoShort-crestedwave interactionwith a con-centric porous cylindrical structurerdquo Applied Ocean Researchvol 29 no 4 pp 199ndash209 2007
[30] L Xiao Y Kou L Tao and L Yang ldquoComparative studyof hydrodynamic performances of breakwaters with double-layered perforated walls attached to ring-shaped very largefloating structuresrdquo Ocean Engineering vol 111 pp 279ndash2912016
[31] T L Yip and A T Chwang ldquoWater wave control by submergedpitching porous platerdquo Journal of Engineering Mechanics vol124 no 4 pp 428ndash434 1998
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
