Wave interactions with double slotted barriers

Michael Isaacson*, John Baldwin, Sundarlingam Premasiri, Gang Yang

Department of Civil Engineering, University of British Columbia, Vancouver, B.C., Canada V6T 1Z4

Received for publication 4 December, 1998

Abstract

The present article outlines the numerical calculation of wave interactions with a pair of thin vertical slotted barriers extending from thewater surface to some distance above the seabed, and describes laboratory tests undertaken to assess the numerical model. The numericalmodel is based on an eigenfunction expansion method and utilizes a boundary condition at the surface of each barrier which accounts forenergy dissipation within the barrier. Comparisons with experimental measurements of the transmission, reflection, and energy dissipationcoefficients for partially submerged slotted barriers show excellent agreement and indicate that the numerical method is able to adequatelyaccount for the energy dissipation by the barriers.q 1999 Elsevier Science Ltd. All rights reserved.

Keywords:Breakwaters; Coastal engineering; Coastal structures; Permeability; Wave reflection; Wave transmission; Waves

1. Introduction

Breakwaters are widely used to provide economicalprotection from waves in harbours and marinas. In certainsituations, breakwaters in the form of thin, rigid, pile-supported vertical barriers which extend some distancedown from the water surface have been used or considered.These have the advantages of allowing water circulation,fish passage and sediment transport beneath the breakwater,and may be relatively economical by providing protectioncloser to the water surface where wave action is mostpronounced. Predictions of wave interactions with suchstructures have been obtained previously by a number ofauthors for the case of an impermeable barrier on thebasis of linear wave diffraction theory. Numerical solutionshave been developed on the basis of the boundary elementmethod (Liu and Abbaspour [11]; Nakamura [16]) and theeigenfunction expansion method (Losada et al. [12]; Abul-Azm [1]).

In some instances, a permeable barrier, such as a slottedvertical barrier made from timber planks, may be preferred.For example, this may be selected in an effort to reduceunwanted wave reflections on the upwave side of the barrier.Thus, the prediction of wave interactions with a permeableor slotted thin vertical barrier is also of interest. A primaryfeature of such interactions is that wave energy is absorbedwithin the structure. Several authors have considered suchpredictions for permeable structures that extend from the

free surface to the seabed (e.g. Sollitt and Cross [19]; Hagi-wara [5]; Bennet et al. [2]; Yu [26]). Most formulationsassume the velocity through the porous medium is propor-tional to the pressure gradient, with a complex proportion-ality constant that accounts for possible phase differencesbetween the velocity and pressure gradient. This descriptioncan be related to the physics of the flow within the structureon the basis of both frictional and inertial effects. Compar-isons with experimental measurements of transmission andreflection coefficients have been carried out for slottedbarriers extending to the seabed by Hagiwara [5]; Kriebel[10]; and Bennet et al. [2]) and generally exhibit satisfactoryagreement. The extension to the case of a partiallysubmerged barrier has been considered more recently byIsaacson et al. [7].

Permeable barriers have the advantage of reducing wavereflection on the upwave side of the barrier but in order toalso reduce wave transmission to an acceptable level it isoften necessary to use two vertical barriers in many practicalapplications. A number of authors have considered theproblem of two or more barriers. These include severalstudies on using multiple porous plates as wave absorbers,e.g. Twu and Lin [22,23] and Losada et al. [13]; experimen-tal studies of a double screen breakwater by Gardner et al.[4] and comparisons between theoretical and experimentalresults for wave reflection and transmission by double verti-cal slotted barriers, e.g. Kondo [9] and Hagiwara [5]. All ofthe above studies have been carried out for barriers extend-ing from the free surface to the seabed, and little work hasbeen reported on double barriers that are partiallysubmerged.

Applied Ocean Research 21 (1999) 81–91

0141-1187/99/$ - see front matterq 1999 Elsevier Science Ltd. All rights reserved.PII: S0141-1187(98)00039-X

* Corresponding author. Tel.: 001 604 822 6412; Fax: 001 604 822 7006;e-mail: isaacson@apsc.ubc.ca

In the present article an eigenfunction method is devel-oped for wave interactions with a pair of thin permeablebarriers extending from the water surface to some distanceabove the seabed. This is an extension of a method devel-oped by Isaacson et al. [7] for a single partially submergedvertical permeable barrier. The method is used to develop anumerical model and theoretical predictions are comparedwith the results from laboratory tests undertaken to assessthe numerical model.

2. Theoretical formulation

2.1. Governing equations

A normally incident, regular, small amplitude wave trainof heightH and angular frequencyv propagates in water ofconstant depthd past two identical thin permeable verticalbarriers a distance 2l apart, as shown in Fig. 1. The barriersextends downwards a distanceh below the still water level.A Cartesian coordinate system (x, z) is defined with xmeasured in the direction of wave propagation from apoint mid-way between the barriers, andz measuredupwards from the seabed. The fluid is assumed incompres-sible and inviscid, and the flow irrotational. The fluidmotion can therefore be described by a velocity potentialF which satisfies the Laplace equation within the fluidregion. In addition, the wave height is assumed sufficientlysmall for linear wave theory to apply. Consequently,F issubject to the usual boundary conditions, linearized whereappropriate, at the seabed, free surface and far field (see, forexample, Sarpkaya and Isaacson [18]), and may thus beexpressed, using complex notation, in the form:

F�x; z; t� � Re�Cf�x; z�exp�2ivt��; �1�where

C � 2igH2v

� �1

cosh�kd� : �2�

Also, Re[ ] denotes that the real part of the argument,i �

����21p

; t is time,k is the wave number, andg is the gravita-tional constant.

The fluid domain is subdivided into three regions by theplanes of the barriers, as shown in Fig. 1, and the two-dimensional potentialf in Eq. (1) is denotedf1, f2 andf3 in regions 1, 2 and 3 respectively. Alongx� ^l, thepressure (and hence the velocity potential) and the horizon-tal velocity are equated along the matching boundarieswithin the fluid, and are applied to a suitable boundarycondition along the surface of each barrier. The conditionsalong the matching boundaries are thus:

f1 � f2;2f1

2x� 2f2

2xalong x� 2l for 0 # z # a;

�3�

f2 � f3;2f2

2x� 2f2

2xalong x� 1l for 0 # z # a;

�4�wherea� d 2 h (see Fig. 1).

2.2. Permeable boundary condition

The boundary condition along the permeable barriersmay be developed on the basis of the formulation of Sollittand Cross [19] (see also Sulisz [21]), and as adopted by Yu[26] for a thin vertical barrier extending to the seabed. Thismay be expressed as:

2f1

2x� 2f2

2x� 2iG 0�f2 2 f1�

along x� 2l for a # z # d;

�5�

2f2

2x� 2f3

2x� 2iG 0�f3 2 f2�

along x� 1l for a # z # d;

�6�

whereG0 � G=b, b is the barrier thickness andG is a perme-ability parameter which is generally complex. Eqs. (5) and(6) correspond to the fluid velocity normal to the barrierbeing proportional to the pressure difference across thebarrier, with a complex constant of proportionality so thatthe real part ofG corresponds to the resistance of the barrierand the imaginary part ofG corresponds to the phase differ-ences between the velocity and the pressure because ofinertial effects. Various authors have related the permeabil-ity parameter,G, to the physics of the flow within the barrierin different ways. For example the resistance may beexpressed in terms of a friction coefficient (e.g. Sollitt andCross [19]), a drag coefficient (e.g. Hagiwara [5]) or a headloss coefficient (e.g. Mei et al. [15]), while the inertialeffects are generally expressed in terms of an added masscoefficient (e.g. Sollitt and Cross [19]) or an effective orificelength (e.g. Mei et al. [15]). In the present article, themethod of Sollitt and Cross [19] is followed andG is

M. Isaacson et al. / Applied Ocean Research 21 (1999) 81–9182

Fig. 1. Definition sketch.

expressed by:

G� 1

f 2 is; �7�

wheree is the porosity of the barrier (defined as the fractionof area occupied by the slots,c=�w 1 c� see Fig. 2),f is afriction coefficient ands is an inertia coefficient given by

s� 1 1 Cm1 2 1

1

� �: �8�

In Eq. (8),Cm is an added mass coefficient. It is noted thatalthough the barrier is taken to have zero thickness withrespect to the wave diffraction problem under consideration,the barrier width is considered non-zero with respect to theflow within the barrier. The friction coefficient,f, comesfrom a linearization of the velocity squared term associatedwith the head loss across the barrier. In the original formu-lation of Sollitt and Cross [19]f is calculated implicitlyusing the Lorentz principle of equivalent work so that thenonlinear effects of wave steepness are retained. Thisrequires an iterative procedure, and in the present articlethe formulation of Yu [26] is followed such thatf is treatedsimply as a constant which is assumed to be known.

2.3. Eigenfunction expansion

Expressions forf1, f2 andf3 which satisfy the seabed,free surface, and radiation conditions, as well as the aboveconditions alongx� ^l, may be developed in terms of foursets of coefficientsA1m, A2m, A3m, andA4m which are initiallyunknown:

f1 � fw 1X∞m�0

A1m cos�mmz�exp�mm�x 1 l�� �x # 2l�; �9�

f2 �X∞m�0

A2mcos�mmz�exp�2mm�x 1 l��

1X∞m�0

A2m cos�mmz�exp�mm�x 2 l��

�2l # x # 1l�;

�10�

f3 �X∞m�0

A4m cos�mmz�exp�2mm�x 2 l�� �x $ 1l�: �11�

Here fw represents the incident wave potential which isgiven as:

fw � cosh�kz�exp�ikx�: �12�Also,mm for m $ 1 are the positive real roots of the follow-ing equation, taken in ascending order:

mmd tan�mmd� � 2v2dg

for m $ 1: �13�

m0 itself corresponds to the imaginary root of the aboveequation, such thatm0 � 2ik, with the wave numberkbeing given as the real root of the corresponding equation:

kd tanh�kd� � v2dg

: �14�

Thus, Eqs. (9)–(11) each represent the incident wave traincombined with a superposition of a propagating mode(m� 0) and a series of evanescent modes (m $ 1) whichdecay with distance away from the barrier. They satisfy allthe relevant boundary conditions, except that the conditionsof pressure continuity along the matching boundary and theboundary condition at the barrier surface are still needed todetermine the coefficientsA1m, A2m, A3m, andA4m.

For 0# z # a, the matching conditions expressed in Eqs.(3) and (4) give rise to the following set of equations forA1m,A2m, A3m, andA4m:

X∞m�0

A1m cos�mmz�2X∞m�0

A2m cos�mmz�

2X∞m�0

A3m cos�mmz�exp�22mml�

� 2cos�m0z�exp�m0l�; �15�

X∞m�0

A1mmm cos�mmz�2X∞m�0

A2mmm cos�mmz�

2X∞m�0

A3mmm cos�mmz�exp�22mml�

� m0 cos�m0z�exp�m0l�; �16�

X∞m�0

A2m cos�mmz�exp�22mml�1X∞m�0

A3 cos�mmz�

2X∞m�0

A4m cos�mmz�

� 0; �17�

M. Isaacson et al. / Applied Ocean Research 21 (1999) 81–91 83

Fig. 2. Geometry of slotted barrier.

X∞m�0

A2mmm cos�mmz�exp�22mml�2X∞m�0

A3mmm cos�mmz�

2X∞m�0

A4mmm cos�mmz�

� 0;

�18�And along the barrier’s surface the boundary conditions(expressed by Eqs. (5) and (6)), give rise to the followingequations fora # z # d:X∞m�0

A1m cos�mmz��mmd 2 iG 0d�1 iG 0dX∞m�0

A2m cos�mmz�

1 iG 0dX∞m�0

A3m cos�mmz�exp�22mml�

� �m0d 1 iG 0d�cos�m0z�exp�m0l�;�19�

iG 0dX∞m�0

A1m cos�mmz�1X∞m�0

A2m cos�mmz��mmd 2 iG 0d�

2X∞m�0

A3m cos�mmz�exp�22mml��mmd 1 iG 0d�

� 2iG 0d cos�m0z�exp�m0l�;�20�

X∞m�0

A2m cos�mmz�exp�22mml��mmd 1 iG 0d�

2X∞m�0

A3m cos�mmz��mmd 2 iG 0d�2 iG 0dX∞m�0

A4m cos�mmz�

� 0;

�21�

iG 0dX∞m�0

A2m cos�mmz�exp�22mml�1 iG 0dX∞m�0

A3m cos�mmz�

1X∞m�0

A4m cos�mmz��mmd 2 iG 0d�

� 0;

�22�Eqs. (15)–(22) are integrated in an appropriate way and thenadded to develop a suitable matrix equation forAim Thus,each equation is first multiplied by cos(mnz), then integratedwith respect tozover the appropriate domain ofz (i.e., fromz� 0 to a, or fromz� a to d), and each pair of resultingequations is then added to obtain four sets of equations for

A im. This gives rise to a matrix equation for A1m, A2m, A3m

and A4m:X∞m�0

C�mn�11

X∞m�0

C�mn�12

X∞m�0

C�mn�13

X∞m�0

C�mn�14

X∞m�0

C�mn�21

X∞m�0

C�mn�22

X∞m�0

C�mn�23

X∞m�0

C�mn�24

X∞m�0

C�mn�31

X∞m�0

C�mn�32

X∞m�0

C�mn�33

X∞m�0

C�mn�34

X∞m�0

C�mn�41

X∞m�0

C�mn�42

X∞m�0

C�mn�43

X∞m�0

C�mn�44

26666666666666666664

37777777777777777775

A1m

A2m

A3m

A4m

8>>>>><>>>>>:

9>>>>>=>>>>>;

�

b1n

b2n

b3n

b4n

8>>>>><>>>>>:

9>>>>>=>>>>>;;

�23�where

C�mn�11 � fmn�0;a�1 �mmd 2 iG 0d�fmn�a;d�; �24�

C�mn�12 � 2fmn�0;a�1 iG 0dfmn�a;d�; �25�

C�mn�13 � exp�22mml��2fmn�0; a�1 iG 0dfmn�a; d�; �26�

C�mn�21 � mmdfmn�0;a�1 iG 0dfmn�a;d�; �27�

C�mn�22 � mmdfmn�0;a�1 �mmd 2 iG 0d�fmn�a; d�; �28�

C�mn�23 � exp�22mml��2mmdfmn�0;a�2 �mmd

1 iG 0d�fmn�a;d��; �29�

C�mn�32 � exp�22mml��fmn�0;a�1 �mmd 1 iG 0d�fmn�a; d��;

�30�

C�mn�33 � fmn�0;a�2 �mmd 2 iG 0d�fmn�a;d�; �31�

C�mn�34 � 2fmn�0;a�2 iG 0dfmn�a;d�; �32�

C�mn�42 � exp�22mml��mmdfmn�0; a�1 iG 0d�fmn�a; d��; �33�

C�mn�43 � 2mmdfmn�0; a�1 iG 0dfmn�a;d�; �34�

C�mn�44 � 2mmdfmn�0; a�1 �mmd 2 iG 0d�fmn�a;d�; �35�

C�mn�14 � C�mn�

24 � C�mn�31 � C9mn�

41 � 0; �36�

b1n � exp�m0l��2f0n�0;a�1 �m0d 1 iG 0d�f0n�a;d��; �37�

b2n � exp�m0l��m0df0n�0;a�2 iG 0df0n�a;d��; �38�

M. Isaacson et al. / Applied Ocean Research 21 (1999) 81–9184

b3n � b4n; �39�

fmn�a;b� �Zb

acos�mmz�cos�mnz�dz

� 12

sin�mz�m

1sin�m1z�

m1

� �ba

for m ± n

and1

4mm�2mmz1 sin�2mmz��ba for m� n:

�40�Also m2 � mm2 mn, andm1 � mm 1 mn.

2.4. Numerical solution

In a numerical solution to the problem, Eq. (23) is trun-cated to a finite number of termsN, and thus becomes acomplex matrix equation of rank 4N which can be solvedfor the firstN unknown values of each set of coefficientsA1m,A2m, A3m and A4m. Once these have been calculated, thevarious quantities of engineering interest may readily beobtained. The method is described in more detail by Yang[25]. If required, it may readily be extended in the usual wayto the cases of oblique waves and/or irregular waves (e.g.Dalrymple et al. [3], Losada et al. [12,14]).

The (real) transmission and reflection coefficients,denotedKt andKr respectively, are defined as the appropri-ate ratios of wave heights:Kt � Ht=H andKr � Hr=H, whereHt and Hr are the transmitted and reflected wave heightsrespectively. These are given in terms ofA1m andA4m by:

Kt � A40j j; �41�

Kr � A10j j: �42�From considerations of energy conservation, these arerelated to the energy dissipation coefficientKe:

K2r 1 K2

t 1 Ke � 1; �43�whereKe is the proportion of the incident wave energy fluxthat is dissipated by the barrier.

Finally, the runupR on the upwave side of the firstbarrier, and the maximum horizontal force per unit widthon each barrier,F can be obtained as:

R� H2

exp�m0l�11

cosh�kd�X∞m�0

A1m cos�mmd������

����� �44�

F1 � rgH2 cosh�kd�

����� X∞m�0

A1mdm 2X∞m�0

A2mdm

2X∞m�0

A3mdm exp�22mml�1 d0 exp�m0l������ �45�

F2 � rgH2 cosh�kd�

����� X∞m�0

A2mdm exp�22mml�1X∞m�0

A3mdm

2X∞m�0

A4mdm

�����;�46�

where

dm � sin�mmd�2 sin�mma�mm

: �47�

3. Description of experiments

Experiments were carried out in the wave flume of theHydraulics Laboratory of the Department of Civil Engineer-ing at the University of British Columbia. The flume is 20 mlong and 0.62 m wide. Waves are generated by a singlepaddle wave actuator located at the upwave end. An artifi-cial beach which is covered by a mat of synthetic hair islocated at the downstream end of the flume in order tominimize wave reflection. The vertical barriers to be testedwere placed in the test section 10 m from the wave genera-tor. Fig. 3 shows a general view of the experimental setup.The permeable wave barriers are constructed of verticalpanels of width w� 2.0 cm, and thickness b� 1.3 cm,such that the porosity of the barriers can be varied by chan-ging the dimensions of the slots between the panel members.

Tests were carried out with a constant water depth of0.45 m and with generator motions corresponding to regularwave trains with five different wave periods(T � 0:6;0:8; 1:0;1:2 and 1.4 s), and specified wave heightscorresponding to a constant wave steepness,H=L � 0:07.Pairs of half and fully immersed barriers (h=d � 0:5 and1.0 respectively) with porosities of 0, 5 and 10% andspacings ofl=d � 0:22;0:55 and 1.10 were tested.

In each test, the water surface elevation was measuredupwave and downwave from the wave barriers using twinwire capacitance wave probes. Three probes were used onthe upwave side of the barriers at distances of 1.0, 1.2 and1.5 m from the upwave barrier, and two probes were used onthe downwave side at distances of 1.0 and 1.2 m from the

M. Isaacson et al. / Applied Ocean Research 21 (1999) 81–91 85

Fig. 3. General view of the wave flume showing the barrier setup.

downwave barrier. These probe positions were selected byconsidering the evanescent wave modes from the barrier,the wave paddle location and the method of reflection analy-sis. Wave generation and data acquisition were controlledby a VAX workstation, using the GEDAP (General Experi-mental control, Data acquisition and Analysis Package)software package developed at the Coastal EngineeringLaboratory of the National Research Council, Canada.The wave probes were sampled at a rate of 20 Hz for aduration of 14 s for each test run.

The wave records obtained from the probes were used toestimate the reflection and transmission coefficients for eachtest. The reflection and transmission coefficient were esti-mated using a least squares method applied to simultaneousmeasurements of the water surface elevation upwave anddownwave of the barriers (e.g. Isaacson [6]). The incidentwave height was obtained from measurements without thebarriers in place. For a further description of the experi-ments, see Premasiri [17].

4. Results and discussion

4.1. Comparison with previous predictions

The numerical method was first developed for a singlebarrier and was validated by comparison with previouspredictions for the limiting cases of an impermeable barrierand a permeable barrier extending down to the seabed. Thiswas described by Isaacson et al. [7]. In the present article themethod is extended to the case of two vertical barriers.Results for the general case of two partially submergedpermeable barriers are not available and the method isvalidated by comparison with previous predictions for the

limiting case of two permeable barriers resting on theseabed (h/d � 1.0). Fig. 4 shows a comparison of thepredicted transmission and reflection coefficients with thenumerical results of Hagiwara [5] as functions of the dimen-sionless spacing between the barriersl /h, for a relativedepthkd� 1:5 and a porositye � 5%. A reasonably closefit between the transmission and reflection coefficientspredicted by the two methods is obtained with a frictioncoefficient of f � 0:5 and an added mass coefficient ofCm � 0:18, despite minor differences in the two formula-tions. Hagiwara [5] relates the resistance component of thepermeability parameter to a drag coefficient rather than thefriction coefficient used here and the resistance term is line-arized in a different way. Also Hagiwara [5] relates theadded mass to an effective orifice length incorrectly sothat the added massCm � 2:19 that he quotes in fact corre-sponds toCm < e. Note that for a comparison to the experi-mental results of Hagiwara [5] for the same case, a better fitis obtained withf � 2:0 andCm � 0:18.

The number of terms used in the eigenfunction expansion

M. Isaacson et al. / Applied Ocean Research 21 (1999) 81–9186

Fig. 4. Comparison of predicted hydrodynamic coefficients with resultsfrom Hagiwara [5] as functions of spacing,lh, for h=d � 1:0, e � 15% andkh� 1:5: (a) transmission coefficient, (b) reflection coefficient. —, presentmethod; and1, Hagiwara [5].

Fig. 5. Comparison between measured and predicted hydrodynamic coeffi-cients as functions ofkh, for h=d � 0:5, e � 5% andl=h� 0:44: (a) trans-mission coefficient, (b) reflection coefficient and (c) energy dissipationcoefficient.O, measured; —, predicted.

has been taken asN � 100. This was found to give accurateresults over the range of values presented here.

4.2. Comparison with experiments

In order to assess further the theoretical model, numericalpredictions were compared with the results of experiments.Figs. 5, 6 and 7 show a comparison of the measured andpredicted transmission, reflection and energy dissipationcoefficients as functions ofkh for a relative drafth=d � 0:5, a porositye � 5% and for barrier spacingsl=h�0:44;1:10 and 2.20 respectively. The figures followexpected trends: the transmission coefficient decreaseswith increasingkh, while the reflection coefficient followsthe opposite trend, and the energy dissipation coefficientKe

is non-zero accounting for the energy loss across the barrier.In general, the numerical model is able to adequately repro-duce the most important features of the experimental results,including the energy dissipation through the slotted barrier,and the measured and predicted hydrodynamic coefficientscompare reasonably well for all three barrier spacings. For

larger relative spacings, peaks in the transmission andreflection coefficients occur when the relative draftkh� np=�2l=h�, corresponding to resonant excitation ofpartial standing waves between the barriers. These wereobserved experimentally and somewhat surprisingly corre-spond to a reduction in the energy dissipation coefficient.

The most notable difference between the numerical andexperimental results occurs in the transmission coefficient:the measured transmission coefficients are consistentlylower than the predicted transmission coefficients at lowerrelative drafts for all three barrier spacings. This is reflectedin higher energy dissipation coefficients for the experimen-tal results and is caused by additional energy losses asso-ciated with vortex formation around the bottom edges of thebarriers. Such losses are generally difficult to estimatebecause of scatter in the experimental results, but are seento be as high as 30%. This is considerably higher than

M. Isaacson et al. / Applied Ocean Research 21 (1999) 81–91 87

Fig. 7. Comparison between measured and predicted hydrodynamic coeffi-cients as functions ofkh, for h=d � 0:5, e � 5% andl=h� 2:2: (a) trans-mission coefficient, (b) reflection coefficient and (c) energy dissipationcoefficient.O, measured; —, predicted.

Fig. 6. Comparison between measured and predicted hydrodynamic coeffi-cients as functions ofkh, for h=d � 0:5, e � 5% andl=h� 1:1: (a) trans-mission coefficient, (b) reflection coefficient and (c) energy dissipationcoefficient.O, measured; —, predicted.

energy losses for single barriers which were previously esti-mated to be between 10% and 20% at lower relative draftsfor permeable [7] and impermeable barriers [8,20]. In thecase of the single barrier the energy losses because of vortexformation are generally small when the transmission coeffi-cient is low and thus are not likely of much importance forpractical applications [7]. In the case of the double barrier,however, the energy losses because of vortex formation arehigher because of the additional barrier and could be signif-icant within the range of some practical applications. Preli-minary efforts to include vortex formation in a numericalmodel are described by Nakamura [16].

In order to apply the numerical model, suitable values ofthe friction and added mass coefficients are needed. It isclear that the values off and Cm, which are empirical bydefinition, depend on the specific conditions of laboratorytests and on the detailed geometry of the slots. An extensiveinvestigation of the variation of these coefficients over a fullrange of experimental conditions and with slot geometrywas not carried out, and is beyond the scope of the presentstudy. For example, their values may differ somewhat fordifferent slot geometries. However, it is expected that thecoefficients will be reasonably constant over a wide range ofconditions, and the purpose of the laboratory tests carriedout here is to assess the numerical model on the basis ofconstant coefficients. Thus, in the present study these coeffi-cients were estimated from previous results with a singlebarrier [7] on the basis of a best fit between the measuredand predicted values of the transmission, reflection andenergy dissipation coefficients. This procedure gives a fric-tion coefficient f � 2:0 and an added mass coefficientCm � 0, which agrees with the values used by Yu [26] forrubblemound breakwaters extending down to the seabed,with the values ofCm used by Twu and Lin [22,23] andLosada et al. [13] for multiple slotted barriers extendingdown to the seabed, and with the values ofCm measuredby Urashima et al. [24] for a single slotted barrier extendingdown to the seabed.

Fig. 8 gives a further comparison between the predictionsof the numerical model and the experimental results, andshows the transmission, reflection and energy dissipationcoefficients as functions of the relative spacing betweenthe barriers forh=d � 0:5 and forkh� 0:95. Overall agree-ment is seen to be satisfactory. It appears that the measuredvariation of the hydrodynamic coefficients with spacing isless than predicted by the numerical model. This effect wasalso observed by Hagiwara [5] for the fully submergedbarrier.

In the design of a slotted breakwater, the choice of poros-ity is particularly important. This is illustrated in Fig. 9which shows the measured and predicted transmission andreflection coefficients as functions of porosity for the halfsubmerged barriers at a relative draftkh� 0:95 and a rela-tive spacingl=h� 1:1. There is some scatter in the experi-mental results but again the overall agreement issatisfactory.

M. Isaacson et al. / Applied Ocean Research 21 (1999) 81–9188

Fig. 8. Comparison between measured and predicted hydrodynamic coeffi-cients as functions of dimensionless spacing,l /h, for h=d � 0:5, e � 5%andkh� 0:95: (a) transmission coefficient, (b) reflection coefficient and (c)energy dissipation coefficient.O, measured; —, predicted.

Fig. 9. Comparison between measured and predicted transmission andreflection coefficients as functions of porositye , for h=d � 0:5, l=h� 1:1andkh� 0:95.O, measured; —, predicted.

4.3. Comparison with single barrier

It is also of interest to compare the performance of adouble barrier with that of a single barrier. Fig. 10 showsa comparison of the transmission, reflection and energydissipation coefficients of single and double barriers as func-tions ofkh, for a relative drafth=d � 0:5, a porositye � 5%,and for double barrier spacingsl=h� 0:44;1:10 and 2.20.As expected, the addition of the second barrier has very littleeffect on the reflection coefficient but gives a noticeabledecrease in the transmission coefficient, because ofincreased energy dissipation by the double barrier. The

influence of barrier spacing can also be seen more clearly:the performance of the double barrier is surprisingly insen-sitive to the spacing between the barriers and is influencedonly by excitation of standing waves between the barriers.

Fig. 11 shows a comparison between the maximum hori-zontal force on both the upwave and downwave barriers andthe maximum horizontal force on a single barrier for abarrier spacingl=h� 0:44 and for h=d � 0:5. For thesame degree of wave protection the wave force on theupwave barrier is less than the force on an equivalent singlebarrier. As expected the force on the downwave barrier isconsiderably smaller.

M. Isaacson et al. / Applied Ocean Research 21 (1999) 81–91 89

Fig. 10. Comparison of hydrodynamic coefficients for single and double barriers as functions ofkh, for h=d � 0:5, l=h� 1:1 ande � 5%: (a) transmissioncoefficient, (b) reflection coefficient and (c) energy dissipation coefficient. —, single barrier; ––– double barrier.

5. Conclusions

The present article describes the numerical prediction ofwave interactions with a pair of thin slotted vertical barriersextending from the water surface to some distance above theseabed. The approach used is based on an eigenfunctionexpansion method and utilizes a boundary condition at thesurface of each permeable barrier which accounts for energydissipation within the barrier. Expressions are developed forparameters of engineering interest, including the transmis-sion and reflection coefficients, the wave runup, and themaximum horizontal force on the barrier.

Comparisons were carried out with previous numericalstudies for a permeable barrier extending down to theseabed, and close agreement was obtained

Laboratory tests were carried out to provide an assess-ment of the numerical model. A comparison of correspond-ing numerical predictions of the transmission, reflection andenergy dissipation coefficients with experimental results isgiven. The agreement is generally satisfactory and indicatesthat the present numerical method is able to adequatelyaccount for the energy dissipation by the slotted break-waters, provided that the relevant empirical coefficientsare suitably chosen. The energy dissipation within theslotted barrier is related to friction and added mass coeffi-cients which have been estimated by fitting with experimen-tal results. A comparison with previous results for a singlebarrier shows that both the transmission coefficient and themaximum horizontal force on the upwave barrier are muchlower for a double barrier, while the reflection coefficient isabout the same.

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