Numerical response of an arbitrarily shaped harbour

Numerical response of an arbitrarily shaped harbour Matiur R a h m a n

Department of Applied Mathematics, Technical University of Nova Scotia, Halifax, Nova Scotia, Canada B3J 2X4 (Received November 1980)

The numerical development of resonance of a harbour of arbitrary shape and depth is studied. The harbour is subdivided into subregions according to the variations of bottom topography such that each subregion is of uniform depth. The Helmholtz wave equation is formulated in each subregion as an integral equation of the Green's theorem. The solution to the entire harbour basin is obtained by a matching procedure at the subregion boundaries. Here, we consider a harbour with basins of constant depths connected in series successively to accommodate a more complicated harbour geometry. An application of this study is made to Kincardine harbour with five basins connected in series successively.

I n t r o d u c t i o n

Harbour osciUation occurs due to waves arriving from the open sea into the harbour. These waves are partly reflected by the boundaries of the harbour and part of the waves is trapped inside the harbour. These waves produce resonance ff the frequencies of the various incident and reflected waves happen to coincide with one or more of the free oscillatory modes of the harbour. Thus it is the concern of the engineers to fred some means of predicting the response of that particular harbour to incident waves.

Recently many theoretical studies have been made on various aspects of harbour oscillations. Most contributions supported by a numerical comparison are confined to harbours with simple and regular geometry; mainly circular and rectangular harbours with a constant depth of water. Both theoretical and experimental studies have been carried out by many early workers including McNown, l Kravtchenko and McNown, 2 Apte and Marcou 3 and Apte a to the case of a circular and rectangular harbours with small and large entrances.

LeMehauteS, 6 has studied the two-dimensional case of resonance in rectangular harbours with various types of entrances connected to an infinitely long but relatively narrow channel. Miles and Munk 7 were the first to treat theoretically the problem of a rectangular harbour connected directly to the open sea. They considered an arbitrary shape harbour and the problem was formulated as an integral equation in terms of a Green's function.

Raichlen and Ippen s investigated the wave-induced oscillations in a rectangular basin. They found a high degree of coupling effect between the small and the highly reflective rectangular wave basin. In order to reduce this coupling effect, efficient wave absorbers and wave filters in the main wave basin were necessary. Miles 9 has shown that the resonant response of a harbour is significantly

affected by the entrance conditions. Knapp and Varoni lo and Wilson n studied the resonance in harbours with more complicated shapes in which they tried to demonstrate the means of minimizing large oscillations in harbours. Wilson et al. 12 studied two- and three-dimensional oscillations in an open basin of variable depth. Raichlen 8 studied two-dimensional oscillations in an irregularly-shaped basin by transforming the equation of continuity and the equations of motion into a set of ordinary differential equations and then solved by the matrix method. Leendertse la studied numerically the oscillations in bays and harbours of irregular shape and variable depth, using a finite difference method with an assumed boundary condition at or near the entrance.

Two independent studies on the oscillations of harbours with arbitrary shape and constant depth have been reported by Hwang and Tuck, 14 Lee is,16 and Lee and Raichlen. 17 Hwang and Tuck obtained their solution by adding the scattered wave potential along the reflecting boundary to the standing wave system.

Recently, Lai and Meiri is studied the wave-induced oscillations in harbours of arbitrary shapes and variable depth. Their study is the extension of the method of Lee is and Lee and Raichlen 17 in order to determine the oscillations in an arbitrary harbour with variable depth.

The paper is concerned with the determination of the oscillations in an arbitrary harbour with variable depth and shape. The study demonstrates the numerical response of a harbour when the entire harbour geometry is discretized according to the bottom topography with uniform basins connected in series successively to accommodate a more complicated harbour.

Govern ing equa t ions

Assuming that the flow is irrotational and the fluid is inviscid and incompressible, the Navier-Stokes equations of

0307-904X/81/020109-13/802.00 © 1981 IPC Business Press Appl Math. Modelling, 1981, Vol 5, April 109

Response of an arbitrarily shaped harbour: M. Rahman

motion for potential flow can be reduced to a simple equation given by the Bernoulli equation:

a¢ 1 . s s s- P + ~ t + 2 t u +v +w)+-p gz=O (11

where ~ is the velocity potential; u, v, w are the velocity components of the fluid particle; p is the pressure; p is the density; g is the acceleration due to gravity; and x, y , z are Cartesian coordinates.

The velocity potential ~(x, y, z, t) satisfies kaplace's equation:

a2~ as¢ as¢ - - + - - + - - = 0 ( 2 ) ax s ay s az a

Boundary conditions The solution to Laplace's equation (2) is subject to the

following boundary conditions:

a~ - - = 0 an

a¢ - - = 0 az

aT/

at

atz = - h

(3)

(4)

a~b at z = 0 (5)

az

1 a¢ 7/ . . . . at z = 0 (6)

g at

where the last two conditions are known as the kinematical and dynamic free surface conditions, respectively. Here, n is the coordinate normal to the boundary, h is the constant depth of the harbour and ~(x,y, t) is the surface elevation.

By the method of separation of variables, the velocity potential $(x, y , z, t) may be expressed as:

I ~(x, y, z, t) = --7- f(x, y) Z(z) e -iot (7)

--1(7

where o = 2n/T, the angular frequency; T is the wave period;f(x, y) is the wave function; and Z(z) is the variation of ¢ in z-direction.

The solutions satisfying the boundary conditions (5), (6) and (7) exist in the following form:

Bo e tch ~(x, y, z, t) = ~ coshk(z + h) f(x, y) e - i ° t

- i o

where Bo is a suitable constant and k is a wave number given by 2nIL in which L is the wavelength.

From the dynamical condition (6) we obtain:

(8)

':0t : [ i°, rl(x'y' t)= - g \~ttlz=o

(9)

in which:

Z(0) = Bo e tch cosh(kh) (10)

and o and k are related through the dispersion relation:

a s = gk tanh(kh) (11)

From the linear shallow water wave theory when the depth is small compared with the wavelength viz, h/L ,~ 1, equation (11) can be approximated as:

a s ~ kSgh (12)

Solution o f Helmholtz equation The Helmholtz equation is

Vsf+ kSf = 0 (13)

where: a s a s

vs_- _ + _ _ ax s ay 2

which has to be solved subject to the following boundary conditions:

(a) a f = 0 along all solid boundaries including (14) an harbour boundary and coastline

./r-r ( af ikf) (b) lim . _ _ - - - = 0 (15) ~ = \ a r

waves radiated from the harbour entrance die out at an infinite distance from the harbour and this is called the radiation condition

The.solution f(x, 2) at any point in the domain of the harbour or outside the harbour can be found in terms of the funetion/(~, ~') at a boundary point (~, ~') and a Green's function G(x,y; ~, ~') which must satisfy the Hehrtho]tz equation.

By applying Green's theorem, we obtain:

f f [/(V~+k s) - G(V s +k2)/] dx dy G

R °:] (16)

S where R is the domain and S represents the distance along the boundary of the domain R.

Here the Green's function G(x, y; ~, ~) is yet unknown. Since:

072 + k 2 ) f = 0

(V 2 + k 2) a = a(x - ~) a c t - ~) (17)

in which 8 is a Dirac delta function. Substituting equation (17) into equation (16), we

obtain:

f :(x, y) a(x - ~) aCY - ~) dx dy

R

= - - - a ds (18) an

s

Now using the property of the generalized function t9 we fred:

- - - G ( 1 9 ) f(~, D = an an J s

where

f f : ( x , y ) a(x - ~) aCv - ~) d,, dy =f(~, ~) (20)

R

a(x - O = o

am a(x- ~)=.o x'-}/i

i fx ¢

and

(21)

1 1 0 App l Math . Model l ing , 1 9 8 1 , V o l 5 , Apr i l

Now writing equation (19) explicitly with respect to the variable arguments:

f ( ~ , D = , y )~a(x ,y ;~ ,D s

a - c (x , y; ~, D ~ I(x, y)] a~(x, y) (22)

Interchanging the roles of (x, y) and (~, ~), namely let (x, y) be the point in the region and (~, ~') be the point on the boundary S and using the symmetrical property of Green's function, equation (22) can be rewritten as follows:

r(x,y) = f i r e , a D ~ a (x ,y ; ~, D

s

a ~-)] ds(~, - C(x, y; ~, D 7n f(~' D (23)

Now we proceed to solve equation (17) which is satisfied by the Green's function.

Writing equation (17) in the polar coordinates, we have:

a2G l a G 1 b2G - - + - - - + - - + k = G = 6 ( r , O ) ( 2 4 ) ar ~ r ar r ~ ao 2 r~=O

in which

a(r, 0) - a(x - ~) a(y - D y - ~ "

tan0 = - - - x - ~

r = [(x - ~)2 + (y - ~)21'~

Without loss of generality we can choose the particular form of Green's function as follows:

G(x, y; ~, D = - 4 HgO(~)

where H(o 1) is a I-Iankel function of first kind and order z e r o .

Substituting equation (26) into equation (23), the function f (x , y ) at the field point (x, y) inside or outside the harbour can be found as a line integral around the boundary of the domain of interest. 2°

(25)

(26)

[(x, y) = - 4 ~) ~ ~<°l~(k') s

' } -H(°O(kr) a-n/(~' ~') ds(~, ~') (27)

Thus to obtain the value of the function f (x , y ) at any point in the domain of interest, we must first know the values of the function f and its normal derivatives af]an on the boundary of the domain.

In order to fred the value of the function f on the boundary, the field point (x, y) is allowed to approach the boundary at a point (~', ~") from the interior of the domain. Thus, when:

(x, y) --* (~', ~')


x , ( f t ) h o = 15'

100 200 300 400 .500 600 700 01 I I I [ I

800

-100

-200

-300

- 4 0 0

- 5 0 0

Delroi~ v

Z

-60(3

-700

- 8 0 0

- 900

-1000

-1t00 T~7

h 4 =9'

K

/ 6

tO

-1300

-1400

Figure I Schematic diagram o f Kincardine Harbour. Prototype scale 1 "~ 100' (1:1200)

equation (27) is modified to give the following.

m',o=- 7 s

r , a D} ds(~ ' -H(o ; (kr ) ~n f (~ , _ ~')

where

r' = [(~' - ~)2 + (~, _ ~)211/2

(28)

(29)

Schemat iza t ion

From the physical point of view, a real harbour is of variable depth and of arbitrary coastline contour. But to predict the characteristics of a harbour it is reasonable to assume that the harbour can be divided into several regions according to its bottom topography. In the present study of Kincar- dine harbour (Figure 1), it is subdivided into five regions of constant but different depths. The velocity potential of each region is assumed to satisfy Laplace's equation together with the boundary conditions. The region outside the harbour, the open sea region, is assumed to be of constant depth and the coastline is taken as perfectly reflecting.

Appl Math. Modelling, 1981, Vol 5, April 111

Response of an arbitrarily shaped harbour: M. Rahman In order to solve equations (27) and (28) we use

numerical integration by dividing the boundary of the harbour into a sufficiently large number of segments of arbitrary size. It should be mentioned that the value of the wave function f on the boundary of each subregion or its normal derivatives are assumed constant for each segment and equal to the value of the function at the midpoint of that segment.

In a general case, in order to write equation (28) in a summation form, we have the following expression:

f ( x i , y 3 = - -

i N - = yj)

) a } aS f%'yi) asj (30)

where (x i , Y i ) is a field point on the boundary; (xi, y / ) is a boundary point; ASj is the length of the jth segment of the boundary;

ri i = [(xi - xj) 2 + (Yi - y j )2] l /2 = r] i (31)

and N is the total number of segments. Equation (30) can be written in the matrix form:

F = bo(GnF - GP) (32)

where

- i b o ~" - -

2

F = f (x i , y i ) i = 1, 2 . . . . . N (33a)

3 P=-~n f (x i , yi) / = 1 , 2 , . . . , N (33b)

(G)i i = H(o l ) (k r i / ) AS i

= {Jo(krij) + iYo(krii)} ASj

for i4=j i , j = 1 , 2 . . . . . N (33c)

where Jo and Yo are respectively the Bessel functions of first and second kind of zeroth order. It is evident that when i =j , i.e. r = 0, Yo(kr) is singular. In order to evaluate the diagonal elements of matrix G we can cite Lee's calcula- tion. is Thus

and

i = 1,2 . . . . . N

(33d)

"l" 3rii (Gn)ij = - k i l t '(kri/) -~n ASi (33e)

i ~ j i , j = 1 , 2 , . . . , N

where H~l)(krij) is the Hankel function of the first kind of order one.

The normal derivative Orij/~n can be expressed in the following difference form:

atilt_ xi--x/(AY)i Yi--y/(Ax t an/ / rii - ~ + ri/ - '~ (33f)

For i = j, the Hankel function Hfl)(kr) is singular. The diagonal elements of matrix G n can be expressed as:

. . i [ a x ~ 2 Y 32x~Y] (Gn)ii----lr ~ss as 2 3s 2 3s ASi (33g)

i = 1 , 2 . . . . . N

Now equation (32) can be arranged as:

(boG n - I ) F = boGP (34)

where I is the identity matrix such that:

0; ff i q:j I = 6ii =

1; i f / = ]

If the inverse of matrix (boG n - I ) exists, we can write the following:

F = (boGn - I ) -1 (boGe) (35)

where the vector P denotes the unknown normal derivatives ofF.

Region I The wave function f l in region I can be determined from

equation (35) by formulating as follows:

F~ = (boGn - I)-l(boGP~) (36)

where

F1 = f l ( x i , Y i ) ; i = 1, 2, . . . , N (37a)

3 P~ = -~nf~(xi, yj); ] = 1, 2 , . . . ,N (37b)

G and Gn are both N~ x N1 matrices which can be computed directly, and N1 is the number of segments into which the boundary of region I is divided, including bl segments for the open sea boundary (AB in Figure 1) between regions I and O; and by the segments for the imaginary boundary CD between regions I and II. The segments of the boundary start at point B and are numbered in the anticlockwise direction.

The unknown values of vector Pa are denoted as (D a)l, ( O 1 ) 2 . . . . , (Da)b, at the open sea boundary AB between regions I and O, and (Dz)m2, . . . . (Dl)n= at the imginary boundary CD between regions I and II, and zero at the solid boundaries. Therefore:

Pl = UIDI (38)

where UI is an NI x BI matrix, D1 is a column vector with B elements and B1 = bl + b2.

Substituting equation (38) into equation (36) and rearranging yields:

F1 = (boGn - 1) -1 (boGUs) D~ = MID1 (39)

where M1 = (boGn - 1) -1 (boGU1) is a known N l x B1 matrix.

Region I I The wave function f2 in region II can be determined from

equation (35) by formulating as follows:

F2 = (boGn - I)-X(boGe2) (40)

where

F2 = f2(xi, Yi); a

e2 = - - [f2(xj, yj)]; On

i = 1 , 2 , . . . , N 2 (41a)

j = l , 2 . . . . . N2 (41b)

112 Appl Math. Modelling, 1981, Vol 5, April

G and Gn are bothN2 x N2 matrices which are known and can be computed, directly;N2 is the number of segments into which the boundary of region II is divided, including b2 segments for the imaginary boundary CD between regions I and II, and b3 segments (ms, m3+1 . . . . , ns) for the imaginary boundary EF between the regions II and III. The segments of the boundary start at point D and are numbered in the counter-clockwise direction.

The unknown values of P2 are denoted as (D2)1, (D2)2, (D2)b= at the imaginary boundary C'~ between regions I an___d II, (D2)m~, (D2)m,+~, (D2)n~ at the imaginary boundary EF between regions II and III, and zero at the solid boundaries.

Thus we have

P2 = U202 (42)

where U2 is anN2 x B2 matrix, B2 = b2 + b3,D2 is a column vector with B2 elements and the values of the normal derivatives at each imaginary boundary are taken in an opposite order.

Substituting equation (42) into equation (40) and rearranging gives:

F2 = (boGn - 1) -1 (boGU2) 0 2 = M2D2 (43)

where M2 = (boGn - I ) - l (boGU2) is an N~ x B2 matrix which can be calculated directly.

Region 111

The wave function fs in region HI can be calculated from equation (35) by formulating as follows:

F3 = (boGn - I ) - I (boGPs) (44)

where

Fs = A ( x i , yi); i = 1, 2 , . . . , N s (45a)

a P3 = -~n [fs(x], Y/)]; / = 1, 2, : . . , Ns (45b)

Here G and G n axe both N3 x Ns matrices which are known and can be computed directly, N 3 is the number of segments into which the boundary of region III is divided, including bs__.segrnents (1, 2 . . . . , bs) for the imaginary boundary EF between regions I1 and III, and b4 segments (m4, m4+l . . . . . n4) for the imaginary boundary G---H between the regions III and W. The segments of the boundary start at the position F and are numbered in a counter- clockwise direction.

The unknown values of the vector Ps are denoted by (Ds)l, (Ds)2 , . . . , (Ds)t,. at the imaginary boundary between regions II and III, (D3~ , , (Ds)m . . . . . - - , (D3)n, at the imaginary boundary GH between the regions III and IV, and zero at the solid boundaries'..Thus we have:

Ps = U~Os (46)

where Us is an N3 x Bs matrix, B3 = bs + b4. D3 is a column vector with Bs elements.

Substituting equation (46) into equation (42) and rearranging yields:

F3 = (boGn - I ) - l (boGUs) D s = MsDs (47)

in which Ms = (boG,, - 1) -1 (boGUs) is an Ns x B3 matrix which can be calculated directly.

Region I V

In the same way as before, the wave function f4 on the boundary of region IV can be formulated as:


F4 = (be Cn - I ) - ' (be GP,) (48)

where

F4=f4(x i , Yi) i = 1, 2 , . . . ,N4 (49a) a

P = - - I f , (x i, y/)] / = 1, 2 . . . . . N, (49b) 0n

G and Gn are both N, x N4 matrices which can be computed directly, N4 is the number of se[0aaents into which the boundary of region IV is divided, including b4 segments (1, 2 . . . . . b4) for the imaginary boundary between regions III and IV and bs segments (ms, ms+l, . . . , ns) for the imaginary boundary between regions IV and V. The segments start at the position H (see Figure 1) and are numbered in a counter-clockwise direction.

The unknown values of vector P4 are denoted as (D,)I, (D4)2, • • •, (D4)b, at the imaginary boundary (if-H) between regions HI and IV, (D4)ra~, ( D * ) m s , t , . . . , (D4)n~ at the imaginary boundary (T~d) between regions IV and V; and zero at the solid boundaries such that:

P, = U, D4 (50)

where U4 is an N4 x B, matrix B4 = b4 + bs , D4 is a column vector with B4 elements and the values of the normal derivatives at each imaginary boundary are taken in an opposite order.

Substituting equation (50) into equation (48) and~rearranging gives:

Fa = (boGn - [)~I(boGU4) D4 = M 4 D , (51)

where M4 is an N4 x B4 matrix which can be calculated directly.

Region V

The wave function fs on the boundary of region V can be written as:

Fs = (boG. - I ) - l (boGPs) (52) where

Fs =fs(x i , Yi) i = 1, 2 . . . . . Ns (53a) a

Ps = - - [fs(xj , yy)] j = 1, 2 . . . . . Ns (53b) an

G and G n are both Ns x Ns matrices, and Ns is the number of segments into which the boundary of region V is divided, including bs segments for the imaginary boundary (TS) between regions IV and V. The segments of the boundary start at point S and are numbered in a counter-clockwise direction.

The unknown values of vector Ps are denoted by (Ds)l, (Ds)~, . . . , (Ds)b~ at the imaginary boundary (TS) between the regions IV and V and zero at the solid boundary. Thus:

Ps = UsDs (54)

where Us is anNs x Bs matrix, Ds is a column vector with bs elements.

Substituting again equation (54) into equation (52) and rearranging yields;

Fs = (boGn - I ) - t (boGUs) Ds = MsDs (55)

where Ms is anNs x Bs matrix which earl be evaluated directly. Further information about the development of the matrix elements can be found in the work of Rahman and Moes. 21


Response of an arbitrarilv shaped harbour: M. Rahman

W a v e f u n e t i o n in the open sea region

The wave function fo in the open sea region can be expressed as a linear approximation of the incident wave, reflected wave and radiated wave. Thus we have:

re(X, y) = J~ (x, y) + fr (x, y) + fs(X, y) (56)

where f i(x, Y) is an incident wave function due to the wave coming from open sea toward shore;fr(x, y ) is a reflected wave function which would occur if the harbour entrance were closed;fs(x, y) a radiated wave function due to the wave being radiated from the harbour entrance. It is also known as the scattered wave function resulting from the presence of the harbour.

The function fs(x, Y) at any point in region 0 can be expressed from Weber's solution of the Helmholtz equation for an external domain as:

A ( x , y ) = - 4 o)

- - o o

_ H(ol)(kr) a 0)1} d~ ~n [fs(~, (57)

in which (~, 0) is a point on the boundary along the x-axis, r the distance between the field point (x, y ) and a boundary point (~, 0) and the integration is carried out along the x-axis. In order to determine the function/s on the boundary of the region 0, the field point is allowed to approach a boundary point (~', 0) and the following expression can be obtained:

i = a ' 0

- H(oD(kor') ~n [fS([, O)] } d ~ (58)

where r' = (~' - ~). The first term on the right hand side of equation (58) is equal to zero because the term

a Z o'(kor ' )

can be written as

- koHO)(ko r') ar~n

but thedistance r' between the two points ~' and ~ on the boundary along the x-axis and n is the outward normal in the y direction, so that ar'/bn = ar'/ay = 0. In the second term, the value of

a

~n f ' (~ ' 0)

is equal to zero along the shoreline, but at the harbour entrance AB this normal derivative is not equal to zero. Thus equation (58) can be simplified to the following ex'pres'sion:

A

i f a fs(~', O) =2 HO)(k°r ) ~n fs(~' O) d~ B

The harbour entrance AB is divided into ba segments start-

(59)

ing at point B, numbered in counter-clockwise direction, and equation (59) can be expressed in a summation form as:

i b, a fs(xi, O) = -2 t~=, H(°l)(k°ri/) ~n [fs(x/, 0)] AS] (60)

where (xi, 0) are the coordinates of the midpoint of segment i at the harbour entrance; (xi, 0) are the coordinates of the midpoint of segment /a t the harbour entrance; AS i is the length of the / th segment of the harbour entrance. Equation (60) can be written in matrix form,

F s = - boHsD s (61)

where,

i

2

F s =fs(Xi , 0) ; i = 1, 2 , . . . , bl (62a) a

Ds = ~n [fs(x/, 0)] i = 1, 2 , . . . , b l (62b)

(n,)o =/t¢o'(korij) ZxSj = [Jo(korq) + iYo(korij)] ASj;

i = 1 , 2 , . . . , b l for i :/=]

] = 1 , 2 , . . . , b l (62c)

i = l , 2 , . . . , b ~ (62d)

At the harbour entrance, y = 0, the incident wave plus the reflected wave is a constant and equal to unity. Thus the function fo at the harbour entrance can be expressed as:

Fo = 1 - b o H s D s = 1 - H D s (63)

in which H = boll s is a bl x bl matrix, and:

17o = fo(Xi, O) i = 1, 2 , . . . , b,

where 1 is a column vector with b i elements equal to unity.

Matching conditions In the previous sections we evaluated the function f at

the boundary for each subregion and the open sea in terms of the unknown normal derivative af/ar at the imaginary boundaries, as well as the harbour entrance. In order to obtain this unknown normal derivative, we have to use the "matching conditions' at the imaginary boundaries and at the harbour entrance.

The continuity conditions at the imaginary boundary (on the harbour entrance) between any two regions i and / at any time can be expressed as follows. The water surface elevation at this boundary must be continuous, i.e.

r/i = r/i (64)

From equation (9):

1 rli = - - Be e kini eosh(kihi) f i(x, y) e - lot

g

Substituting this expression into equation (64) and re-


arranging, we have:

e~lhi cosh (kih~) f i ( x ,Y ) - ~ ~ ( x , y ) (65)

ekihi cosh(kihi)

The second continuity condition is that the mass transport across the imaginary boundary from region i into region/ must be equal and opposite. In terms of velocity potential function, it is given by:

0 0

f a~b i aO/dz ann d z = - f an

(66)

- -h i - -h i

Thus from equation (8):

f _ _ Be e ki hi a 0 a~bi dz = ~ sinh(kihi) e - i° t an - i o k i ~nn (~'(X, y)

- - h i

Substituting the above expression into equation (66) and rearranging yields:

k i ekihi sinh (k/hi) a --A(x,y) = ~n k i e kihi sinh(kihi) an [f/(x,y)] (67)

The final solution can be obtained by reducing the matrix equation in each subregion to express only the wave function at the imaginary boundaries and at the harbour entrance, and using the relationship given in equations (66) and (67).

Region I In order to write equation (39) for the functionf~ at the

harbour entrance AB and the imaginary boundary CD, we define another matrix Mta which consists of the first b rows, and the m~th row to the n~th row of matrix M~. The values of the function/'1 at the harbour entrance AB are denoted by (F~)~, (F~)~,..., (F~)~ and at the imaginary boundary CD by (Fl)mz, (F1)m~+ ~, . . . , (F,)n~, such that:

Fla = Mt4D~ (68)

and where F~a is a column vector with B~ elements, B~ = b ~ + b~;ML~ is a B, x B ~ matrix; and D ~ is a column vector with B1 elements.

Region H The functions f2 at the imaginary boundaries CD and EF

are, respectively, denoted by (F2)b (F2),z , . . . , (F2)b,, and by (F2)m: (F2)m~.: . . . , (F2)n~ such that:

F2A = M2A D2 (69)

where F2a is a column vector with B2 elements, B2 = b2 + b3; M2a is a B: x B2 matrix which consists of the first b~ rows, and the math row to hath row of the matrix M~ in equation (43); D2 is a column vector with B~ elements.

The first b: elements of vectors Fza and D~, are respectively related to the b~ elements of vectors F ta and D~ in equation (68), and at the imaginary boundary CD according to equations (66) and (67). Thus:

[(F?)~ ]= ek'~'cosh(k,h'i ") [ (F , )n , ] (70a)

(F~)b, J ' ,n, cosh(k~h2) x.[(F;)m J

and

[ (D:)~"] =- k' e*'h' s~(k '" ' )x [(Zh)"'] (70b) (D2), J -k-~ e*#',----sinh(k~h:--------~ I_(D,),,,J


Substituting these equations into equation (69), we have:

M r r H r# r ( F ~ ) . 1 [ ( 2 ) l l . . . ( g 2 ) ~ , ~ ( M ~ ) l b . , . . • ( M 2 h . B ~ ]

• ~ . . . . ; . . . '. . . . . . . , . . . . . . . . , ; . : . . . . . . . . . . . ,; . . . . . j

-(D,.)~] x (O,).~|

](D:)n'l (71a)

(D~)m,J in which the first b2 columns of the above matrix, marked with primes, are multiplied by the factor:

k: tanh(klhl)]

" kl tanh(k2h2)J

and the last ba columns of this matrix, marked with double primes, are multiplied by the factor:

[ e k~h~ cosh(k2h2)]

ek, h, cosh(klhl)l

The last b3 equations in equation (69) can be written as the following:

q I . . . . . . . : " . . . . . . . . . . . . . . . . . . . . . . . . . . t (F2)ra 2 )m 3 , . . . ( M 2 )m3.b2 (M2)m3,b,+,... ( 2)ra,,B

(F,}oj L(M; " ) , , . • . ( M , " . . . ( M , ) o , . . , j

(D,)~2 ( D , ) . ~

x (D2)n, (71b)

(O2)ms.

in which the first b2 columns of the matrix marked with triple primes, are multiplied by the factor

-- k 2 e k,h,

kl e %h~ sinh(k2h2) J

Region III The function f3 at the imaginary boundaries EF and GH

are, respectively, denoted by (F3)l, (F3): . . . . . (F~)t,, and by (F3)m,, (F3)m, . , , . . . , (F3)n,, such that:

Fan = MaaDa (72)

where F ~ is a column vector with Ba dements; Ba = b3 + b4; Maa is aB3x Bs matrix which consists of the first b3 rows and the m4th to the n4th row of the matrix M3 in equation (47); and D3 is a column vector with Ba elements.

The first ba elements of the vector Faa and D are, respectively, related to the b3 dements of the vectors F2A and D1 in equation (71b), and at the imaginary boundary EF according to equations (65) and (66)• Thus:

• = e %h2 cosh(k2ha) . (73a)

k(4) d ek'"c° (k3h3)× L( 2i-J

Appl Math• Modelling, 1981, Vol 5, April 115

Response of an arbitrari ly shaped harbour: 114. Rahman i

and

. = x • (73b)

d k ek'"'s (k3h3)

Substituting these two expressions into equation (72) and rearranging the fixst b3 equations we obtain:

(F2!n, (Ms)I,I"" (M3)l,/h(M3)l,b,., -.. ( 3)I.B3

0 tt Mr! (F,;,,,,J

x (D~.)m3l (74a)

_W;).,J where the first b columns of the above matrix, marked with primes, are multiplied by the factor

[ - k3 tanh(k2h2) ] -j

and the last b4 columns, marked with double primes, are multiplied by the factor

[ ek~h cosh(kah3)]

The last b4 equations in equation (72) can be written as follows:

(F3;,. j ,-( M ; )ra , , . . . ( M'3 )m 4.b 3 . . . ( M a )ra,.b ,+, . . . ( M 3 )m ,.B , ]

(Mi)n,:- .- (Mi)n4,b,... (MS)n,,b,,,... (Ua)n4,B3 J

X (D2)m 31 (74b)

(D3)n, J

in which the first b columns of the matrix-in equation (74b), marked with primes, are multiplied by the factor

[-- ks e/%h2 sinh(k2h2) ]

k 2 e/%h~ ~ J

Region IV The function I'4 at the imaginary boundaries GH and TS

are, respectively, denoted by (F4)l, (F4)2, • • •, (F4)b,, and by (F4)m s, (F4)ms . . . . . . (F4)ns such that'

F4A = M4AD4 (75)

where F4A is a column vector with B4 elements: B4 = b4 + b s ; M 4 a is aB4 X B4 matrix which consists of the first b4 rows, and the msth row to nsth row of the matrix )1'/4 in equation (51); and D is a column vector with B4 elements.

The first b4 elements of vectors F4a and D4 are, respectively, related to the b4 elements of the vectors Faa and D3

in equation (74b), and at the imaginary boundary GH according to equations (66) and (67). Thus:

e k'h" cosh(k4h4) x (76a) k(F4)b,J k(Fa')md

(D.4)b = -k4 e k'h' sinh(kah3) (76b)

• k3 e k4h, sinh(k4h4) "

L (")' ] Substituting equations (76a) and (76b) into equation (75) and rearranging, the first ba equations in equation (75) can be written as the following:

(F3!n, -(M4)I, 1 (M;)l,b4 ~ 4)1,b4÷, (M4)I,B,

L(Fs)mJ 5Mgb,,, (MDb,.b, (M:)b,.b,÷, (MD~,,~,J (O3)m. I (Ds) . . I

X (D4)n. I (77a)

(D4)m, I in which the first ba columns of the above matrix, marked with primes, are multiplied by the factor

k 3 tanh(k4h4) ..I and the last bs columns of this matrix, marked with double primes, are multiplied by the factor

e/¢'h" cosh(k4h4)]

The last bs equations in equation (75) can be written as follows:

(F4!m sl

- III II 41 (M4)ms,. (M4')ms,b4 (M4)ras,b.*, (M4)ms,B

I (M; '%, , , (M;')...b. ( M , ) . , : , . ., (M,),..8.J

(D3)m. (/93),.

x (D4)ns (77b)

(V,)., in which the first b4 columns of the matrix in equation (77b) marked with triple primes, are multiplied by the factor

[_k4e k3hs sinh(kah3)]

k 3 e k4/'14 ~ J

Region V The first bs equations in equation (55) consisting of the

function fs at the imaginary boundary TS are denoted by


(Fs),, (Fsh, (Fs)b,. Thus: FSA = MsADs (78)

where FSA is a column vector with bs elements;MsA is a bs x b5 matrix; and Ds is a column'vector with bs elements.

At the imaginary boundary TS the vectors FsA and Ds are, respectively, related to the last bs elements of vectors F4A and D. given in equation (75) as follows:

(Fs!l]= ek*h*c°sh(k4h'*) [(F4!ns 1 (79a)

.(E,;o,j ek'"'c°sh(kshs)X L(F,;m,j and

.] Ds = = x (D;)bJ k'ek'"'sam(k'h') k(D,).d

Substituting these two expressions into equation (78) and rearranging yields:

(FDm,J [(M;)b,, (M;)o,.b,J L(D,)md in which the elements of the above matrix, marked with primes, are multiplied by the factor

-- ks tanh(k4h4)]

Region 0 (Open sea region) Equation (63) can be written as follows:

F0 = 1 -HD s (81)

in which the values of the function fo at the harbour entrance AB are denoted by (Fo)b (Fo)2, (Fo)l,,, and their normal derivatives are denoted by (Ds)l, (Ds)2, . . . , (Ds)bc The vectors Fo and Ds are, respectively, related to the first bl elements of the vectors F l a and D~ given in equation (68), at the harbour entrance, such that

and

F(Fo),]=ek'h'cosh(k,h,) [(F1.), ] F0= L(Fo) ,j e 0hoco,h(koho) x L(F')bd (82a)

= I (D?' ] - -koek'h'sinh(klhl)[ (DI")' ] (82b)

D, LW;) ,J k'e~°hosinh(k°h°) L(D )o,J Substituting these expressions into equation (81) and rearranging gives:

_(Fi)b,J e~'h' cosh(k,hl)

x (83)

LH~,., n~,.b,J Lw ) ,J In the above equation the matrix H' equals the matrix H in equation (81) multiplied by the factor:

ko tanh(klhl) k ~ tanh ( k oh o )

Response of an arbitrarily shaped harbour: M. Rahman The next step is to put the matrix equations for each

subregion, as well as the open-sea region, into two matrices MA and HA and to solve for the normal derivative of the function f at the imaginary boundaries and the harbour entrance.

Final solution

The matrix equations in regions I, III and V can be com- bined to fred an expression for the function f and its normal derivatives at all the imaginary boundaries and the harbour entrance, such that:

FA = MADA (84)

where F A is a column vector with B elements of the function of f a t all the imaginary boundaries and the harbour entrance:

B=bl +b2 +b3 +b4 +bs MA is a B x B matrix consisting of the matrices MIA, M~A and M~A. DA is a column vector with B elements of the normal derivative of f a t all imaginary boundaries and the harbour entrance.

In order to obtain an expression similar to equation (84), we can combine the matrices arising in region O, region. II and region IV into the following form; such that:

FA = CA + HADA (85)

where F A and DA are both column vectors with B elements as defined earlier. Ca is a column vector with B elements, its first b i elements are

e k°h° cosh(koho)

e k, h, cosh(klh 1)

and other estimates are zero.

Open sea - 0 h o

x ,6 . . . . 1

Region I F ; I ~ hi A+

2

Reg)onh2 II ~>

24 k U1: ~

I ~ j*

H +

ha

Region ]3Z h4

22

N~ = 43 No of segrr~nts in region I gmn]V.~ N2 = 50 . . . . 1I h5 N3 = 43 TIT

24~ N4 = gO ]~Z N 5 = 48

Figure 2 Numerical model of Kincardine Harbour

Appl Math, Modelling, 1981, Vol 5, April 117


• " t .

Ha is a B x B matrix consisting of matnces H m region t O, Mza in region II and M,~a in region IV.

By equating equations (84) and (85), one can obtain the following matrix equation!

MaDa=Ca + HaDa

Solving for, unknown Da :

Oa = (Ma - HA) -1 CA

(86)

(see Figure 1)• The harbour is divided into five regions according to its bottom topography. Each subregion is of uniform depth. The method for an arbitrarily shaped harbour with variable depth is used to calculate the res- ponses inside the harbour.

For the present method, the depth is assumed constant for each subregion. The boundary of subregion I is divided

(87) 4

where (Ma - HA) -1 is the inverse of the matrix (Ma - HA), which is known, and CA is a known vector•

Finally vector Da which consists of values of the normal Q: 3

derivative of the wave function at all the imaginary boun- b" daries and the harbour entrance, can be evaluated from equation (87). g 2

Once we know vector DA, the wave function f at the boundary of each subregion can be determined. The func- :~ tion f l at the boundary region I can be determined from ~ 1 equation (39)• Similarly, functions f2, fs,/'4 and fs at the < boundary of regions II, III, IV and V respectively, can be evaluated from equations (43), (47), (51) and (55).

Now we are ready to evaluate wave function f at any position inside the harbour from equation (27) which can be rewritten in summation form and applied to each one of the subregions inside the harbour. Function f~ at any point (x, y) inside region I can be evaluated from the following expression: 4

f ' (x 'Y)= --4/~1"= ,(xi, Yi) - k , H 1)(kl, r ) 3r ~t_. 3

o

-H(oZ)(kl, r)-~nfl(xi, y i) dS i (88/ 2

All the terms on the right hand side are known, and have been previously deemed. The functions f2, f3, f4 and fs at ~ 1 any point (x, y) inside regions II, III, IV and V respec- < t ive ly can be formula ted exact ly the same way as was int roduced for El.

A m p l i f i c a t i o n factor

The amplification factor R is defined as the ratio between the output and the input of a forced system. In harbour oscillations the amplification factor at any point (x, y) 6 inside the harbour is d&med as the ratio of the wave ampli- tude at point (x,y) to the sum of the incident and reflected waves at the coastline with entrance closed. 5

Thus the amplification factor in the region I is deemed a s

4 Inl(x, y , t) l _ Z1(0) j

R 1 - - - I A ( x , y ) l (89) o Ino(X, 0, t)] Zo(0)

3 In the same manner the amplification factors in the regions o_ = II, III, IV and V can be expressed by:

Z.,(O) ~. 2 Rm Zo(0) I fm(x ,y ) l (90) <E

where m = 2, 3, 4, 5. The absolute value of f is taken 1 because f i s a complex function.

Application, Kincardine harbour

A theoretical investigation is carried out for wave-induced oscillations in a mathematical model of Kincardine harbour

o

Figure 3

0.010 0 .020 0 030 0 0 4 0 0.05( Fnzqu¢tncy 0, ( H z )

Numerical response o f K incard ine Harbour• Region I, Stat ion A

0 0

Figure 4 Stat ion B

] I I I I I l I I 0.010 0 .020 0030 0 . 0 4 0 0 .050

Fnzqu¢ncy 0", ( H z )

Numerical response o f K incard ine Harbour . Region I,

C 0

Figure 5 Stat ion C

I

0 010 0 .020 0.030 Q 0 4 0 0 0 5 0 Fr~zquency G, ( Hz )

Numerical response o f K incard ine Harbour• Region II,


8

a: 5

L O

O 4

g

3 E <

0 0

Figure 6 Station D

I I I I I I I I I I 0.o10 0 0 2 0 0 .030 0 0 4 0 0 .050

Freo~Jency o, ( H z )

Numer ical response o f K incard ine Harbour . Region II,

L o u

£z

E <

Response o f an arbitrar i ly shaped harbour: M. Rahman

7 1

0 0

Figure 8 S t a t i o n F

I I I I I I I I I I 0.010 0 . 0 2 0 0 0 3 0 0.040 0 .050

Frequency o, (Hz)

Numerical response o f K incard ine Harbour. Region I'11,

4

3

E 2 <

0 0

Figure 7 Stat ion E

0.010 0.020 0 .030 0 .040 0 0 5 0 • Frequency O , ( H z )

Numer ical response o f K incard ine Harbour . Region I I I ,

into 43 unequal segments including.6 segments for the imaginary boundary between regions I and II. The boundary of subregion II is divided into 50 unequal segments including 6 segments for the imaginary boundary between subregions II and III. The boundary of subregion III is divided into 43 unequal segments including 6 segments for the imaginary boundary between subregions II and III, and 6 segments for the imaginary boundary between subregions III and IV. The boundary of subregion IV is divided into 90 unequal segments including 6 segments for the imaginary boundary between subregions III and IV, and 6 segments for the imaginary boundary between subregions IV and V. The boundary of

10

Q: 6

o

c 5 .O

~. 4 - E

3 -

2 -

1 -

0 0

Figure 9 Stat ion G

I I ' - " r I I I I I ; 0 0 1 0 o. 020 0 .030 Q 0 4 0 0 0 5 0

Frequency c r ( Hz )

Numerical response o f K incard ine Hart)our. Region IV,

A p p l M a t h . M o d e l l i n g , 1 9 8 1 , V o l 5, A p r i l 1 1 9

Response o f an arbitrari ly shaped harbour: M. Rahman

lO

g

C 6 0

o

E 4 <

0 -

o

10

6

t_ O

5 E O

4 E <

0 0

0.010 0 .020 0.030 0 .040 0.050 Fr'cquency 0", ( Hz )

Figure 10 Numerical response of Kincardine Harbour. Region IV, Station H 9

Figure 11 Station I

0.010 0.020 0.030 Frequency O, (Hz)

0.040 0.050

Numerical response of Kincardine Harbour. Region IV,

subregion V is divided into 48 unequal segments. The segments in each subregion are numbered counterclockwise as is shown in Figure 2.

8

Numerical response Response curves at ten different points A+, B+, C+, D+,

E+, F+, G+, H+, I+, and J÷ (see Figure 2) inside the harbour 6 are presented in Figures 3-12 for the following.

The depth of the water outside the harbour, ho, is con- Q: stant and equal to 5.6 m (18.4 ft). Inside the harbour, the ~ 5 water depth varies in the following manner: h~ = 5.3 m (17.4 ft) in subregion I;h2 = 4 m (13.2 ft) in subregion II; h3 = 4.4 m (14.5 ft) in subregion iii; h4 = 3.3 m (10.8 ft) in o 4

O

subregion IV; and hs = 1.9 m (6.2 ft) in V. u For all the response curves, the ordinate is the arnplifica-

tion factor R, as defined earlier, and the abscissa is the wave ~ 3 frequency in I-Iz. It is to be noted that the number of segments in which each subregion is divided must be significant. Lee Is has demonstrated that accurate results could be 2

achieved if the condition AS/L <~ 1/10 is satisfied in deter- mining the segment size AS, where L represents the wavelength. This condition was employed in the present calcula- tion.

Figures 3-12 display the various resonance modes inside the harbour. In the range of 0.005-0.05 Hz, it has been found that the harbour experiences a few resonance modes of different amplification factors. The numerical response indicates that as the waves approach the main basin (e.g.

1 m

o o

Figure 12 Station J

0.010 0.020 0.030 0.040 0.050 Frequency o, ( Hz )

Numerical response of Kincardine Harbour, Region IV,


subregion IV), the amplification factor R becomes large at 0.0075 Hz. From Figures 9-11, one can observe that the amplification factor R becomes large at 0.038 Hz.

References

1 McNown, J. S. "Waves and seiche in idealized ports', Gravity Wave Syrap., National Bureau of Standards, Cir. 521, 1952

2 Kxavtehenko, J. and MeNown, J. S. Q. Appl. Math. 1955, 13, 19

3 Apte, A. S. and Marcou, C. 'Seiche in ports', 5th Conf. Coastal Eng., Grenoble, France, 1954, pp. 85-94

4 Apte, A. S. 'Recherches theoriques et experimentales sur les mouvements des liquids pesants avec surface fibre', Publ. Sci. et Tech. du Ministere de L'air, 1957, No. 333.

5 LeMeaute, B. J. HydraulicsDiv., ASCE 1960, 86, (Hy9), Proc. paper 2646

6 LeMeaute, B.J. HydraulicDiv.,ASCE1961,87,(Hy2),31 7 Miles, J. and Munk, W. J. Waterways Harbours Div., ASCE

1961,87, (WW3), Proc. paper 2888, 111 8 Raichlen, F. and Ippen, A. T.J. HydraulicsDiv.,ASCE 1965,

91, (Hy2), 1 9 Miles, J.J. FluidMech. 1971,46, Part 2, 241

10 Knapp, R. T. and Varoni, V. A. 'Wave and surge study for the naval operating base, Terminal Island, California', Hydraulic Structures Laboratory of the California Institute of Technology, 1945, Pasadena, California

Response of an arbitrari/y shaped harbour: M. Rahman

11 Wilson, B. W. 'Research and model studies on wave action in Table Bay Harbour, Capetown', Trans. South African Instn Civ. Eng. 1959, 1, (6),

12 Leendertse, J. J. 'Aspects of a computational model for study for a surge-action model of Monterey Harbour, California', Rep. 2-I 36, Science Engineering Associates, San Marino, California, 1965

13 Wilson, B. W. et al. 'Feasibility long period waves propaga- tion', Memorandum RM-5294, PR, 1967, The Rand Corpora- tion, Santa Monica, California

14 Hwang, L. S. and Tuck, E. O. J. FluidMech. 1970, 42,447 15 Lee, J. J. 'Wave induced oscillations in harbours of arbitrary

shape', Rep. KH-R-21, 1969, W. M. Keck Laboratory of Hydraulics and Water Resources, California Institute of Tech- nology, Pasadena, California

16 Lee, J.J.J. FluidMech. 1971,45,375 17 Lee, J. J. and Raichlen, F. "Wave induced oscillations in harbour~

with connected basins', Rep. KH-R-26, 1971, W. M. Keck Laboratory of Hydraulics and Water Resources, California Institute of Technology, Pasadena, California. Waterways, Har bours and Coastal Eng. Die., A SCE 1972, 98, (WW3), 311

18 Lai, R. Y. and Meiri, D. 'Wave induced oscillations in harbours of arbitrary shape and of variable depth', Special report No. 34, 1977, Centre for Great Lakes Studies, The University of Wisconsin, Milwaukee

19 Lighthill, M. J. 'Introduction to Fourier analysis and generalized functions', Cambridge University Press, Cambridge, 1964

20 Baker, B. B. and Copson, E. T. "The mathematical theory of Huygens' principle', Clarendon Press, Oxford (2nd Ed),;t950, pp. 23-28

21 Rahman, M. and Moes, J. Hydraulics Laboratory, National Research Council of Canada, Laboratory Memo. HY-203, 1979


Numerical response of an arbitrarily shaped harbour

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