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Diffraction through wide submerged breakwaters under oblique waves
Ocean Engng, Vol. 21, No. 7, pp. 683-706, 1994
~)
Pergamon
Copyright © 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 00"29-8018194 $7.00 + .00
DIFFRACTION THROUGH BREAKWATERS UNDER
WIDE SUBMERGED OBLIQUE WAVES
A. G. ABUL-AZM Irrigation and Hydraulics Department, Faculty of Engineering, Cairo University, Giza, Egypt Abstract--The eigenfunction expansion method is used to develop a theoretical solution for the flow field caused by monochromatic oblique waves incident upon an infinitely long, submerged, wide and vertical breakwater situated in arbitrary water depth. The breakwater is assumed to be uniform in the y-direction, rigid and impermeable, and the fluid motion is idealized as linearized, two-dimensional potential flow. Numerical results are presented to show the effect of different wave and structural parameters on the transmitted and reflected waves and the hydrodynamic loadings on the breakwater. The accuracy of the solution is verified by comparing the numerical results with the available theoretical or experimental results. Two different simplified solutions for the problem are presented under the approximation of planewaves and under the assumption of long oblique waves.
INTRODUCTION
THE MAIN function of a breakwater is to protect the shoreward area in the leeside of the structure by reflecting or breaking the incident waves. Offshore, construction of a conventional type breakwater--which extends above the highest expected sea level-may have some drawbacks, such as the high construction cost and the obstruction of the view, particularly in front of a pleasure beach or a marina. A wide submerged breakwater may be used to overcome these advantages, and if properly designed, it may significantly reduce the wave energy near shore. Recently, submerged breakwaters have been constructed in prone areas to replace coral reefs in controlling beach erosion or as permanent headlands for the formation of crenulate beaches; see Silvester (1976). Wave propagation over various two-dimensional underwater structures has been widely studied. SoUitt and Cross (1972) and Madsen (1983) have considered the dissipation of wave energy inside rectangular, emerged and porous structures under normal wave incidence. Sulisz (1985) solved the problem for arbitrary cross-section and Dalrymple et al. (1991) have utilized the eigenfunction method to show that for oblique waves incident upon a vertical porous structure, the reflection and transmission coefficients are significantly altered. Wave propagation over sloping and submerged breakwaters has also been studied analytically by Kobayashi and Wurtanjo (1989) and Sabuncu and Goren (1992), who presented an approximate method based on representing the smooth slope by a stepwise geometry and matching the boundary conditions at each step. The problem of a rigid, thin, submerged and impermeable barrier, which does not take into consideration the width effect, has been studied theoretically by many investigators, such as Dean (1945) in the case of infinitely deep water, and Johnson 683
684
A, G. ABuL-AZM
et al. (1951), who provided an approximate solution based on equating the incident
and the transmitted wave powers, and presented experimental results for a submerged rectangular breakwater. Recently, Abul-Azm (1993) presented a solution to determine the flow field around different types of thin, rigid and submerged breakwaters. The method used was the eigenfunction technique together with the least-squares method for the solution of the mixed boundary condition. This method is proven to be analytically straightforward and numerically efficient, as shown by Dalrymple and Martin (1990) in predicting the wave reflection for a long linear array of offshore detached breakwaters. The effectiveness of rigid, wide and submerged breakwaters to dissipate wave energy had been studied experimentally by Hall and Hall (1940), Dick and Brebner (1968), Dattatri et al. (1978) and Abdul Khader and Rai (1980). Jeffreys (1944) had presented an approximate analytical expression to compute the wave transmission through wide submerged breakwaters situated in shallow waters. In this paper the eigenfunction expansion method is adopted to estimate the flow potential around a rigid, impermeable, wide rectangular and submerged breakwater subjected to oblique waves. Numerical results for the transmission coefficients are compared with the experimental results of Jeffreys (1944), Dattatri et al. (1978) and Abdul Khader and Rai (1980). Further numerical results are presented to show the effect of different wave and structural parameters on the transmission coefficient, reflection coefficient and the hydrodynamic horizontal and vertical forces on the breakwater. Simplified analytical solutions are presented using the plane-wave approximation and the long wave assumption. Numerical results obtained from the approximation methods are used to illustrate the validity of the solutions. THEORETICAL FORMULATION The geometry of the submerged breakwater under investigation is depicted in Fig. 1. The system is idealized as two-dimensional, and Cartesian coordinates are employed. The breakwater is assumed to be rigid and impermeable, and has a width W and height h above the sea-bed. The z-axis is directed vertically upwards from an origin on the sea-bed in water of arbitrary depth d. The fluid domain is divided into three regions: region (1) for x - 0, region 2 for x --- W and region (3) for 0 < x < W. The breakwater is subjected to an incident train of regular, monochromatic, small amplitude waves of height H and frequency (o, obliquely propagating with an angle 0 measured anticlockwise from the positive x-direction as shown in Fig. 1. The fluid is assumed to be inviscid, incompressible and the motion is irrotational. The velocity potential ( x , y , z , t ) can be expressed as (oH
dpi(x'Y'Z't)Re = ~ o f~i(X'z)ei(k°ysinO--t°t)'
i = 1,2,3
(1)
where Re is the real part of the complex expression and ko is the wave number which is related to the angular frequency through the dispersion relation (o2 = gko tanhkod. Here it should be noted that the y-variation of the solution must be the same in each region (Snell's Law)--see Dalrymple et al. (1991)--in order to later satisfy the matching condition at each vertical interface. The fluid velocity vector is given by qi = V~i; then the following boundary-valueproblem is obtained in each region i = 1,2,3 (Sarpkaya and Isaacson, 1981) as follows:
685
Wide submerged breakwaters at
HI v
HT
HR
r
(1)
(3)
(2)
w
m
X
FIG. 1. Definition sketch.
V2(~i = 0
Oz
- 0
~(I) i
g~
-
o)2(I)i
=
0
everywhere in the region of flow
(2a)
o n z = 0 f o r i = 1,2 and z = h f o r i = 3
(2b)
on z = d
(2c)
where g is the acceleration due to gravity. In subsequent sections, the total velocity potential will be decomposed into incident and scattered potentials, namely ~ = ~z + ~s. At large distances from the breakwater, • s must satisfy a radiation condition, namely
686
A . G . ABuL-AZM
lim ( O - T - i k o ] ( ~ i s ) = O ,
\ox
i= land2
/
where the incident potential
(I) I
(3)
is given by
* , = Zo(z) e ikoc°s°
(4)
and Zo(z) is the depth dependency function. The solution across each interface, that is, x = 0 and x = W, must be continuous and satisfy the continuity of mass flux and pressure; thus it requires that at x = 0: ( I ) l x = (I)3x ,
h -< z -< d
(5a)
~1 = qb3,
h-< z -
(5b)
d
and from the structural boundary conditions:
(5c)
O<-z<-h.
~lx = 0, Similarly at x = W (I)2x = (I)3x ,
h -< z -< d
(6a)
(I~ 2 = (I)3,
h < z -< d
(6b)
and (6c)
O<_z<--h
qb2~=0 ,
where the subscript indicates derivative with respect to x. EIGENFUNCTION EXPANSION SOLUTION An appropriate solution for the potentials which satisfies Equation (2) may be written as: oe
fl)l = Zo(Z ) ei,%~coso + ~
A,,, e ~,,'~ Zm(z)
(7a)
m=O
fl)2 = ~
n m e -~,,,(x-w) Z m ( Z )
(7b)
{Cm e-xmx + D,.,, ex'(x-w)} Qm(z).
(7c)
m=O oo
~3 = ~ m=O
Am, Bin, Cm and Dm, m = 0,1,2,... represent the unknown potential coefficients which are yet to be determined through the interface conditions (5) and (6). In the summation terms:
-ikocosO
~
and
I = (~k~
m= 0 (8a)
+ k~sin20
m ~ 1
Wide submerged breakwaters
•m =
- i X/132- k2sin20
m = 0
,/ 132 + k2sin20
m -> 1
687
(8b)
where km, m -> 1 are the evanescent wave modes in regions (1) and (2), which are determined as the positive real roots of the transcendental equation, oa2+gkm tankmd = O. 13o is the wave number inside region (3), computed as the real root of the dispersion relation (DE -- g~o tanh~o(d-h) = 0 and [~,,, m -> 1 are its associated evanescent modes obtained from the expression (D2+g[3,,, tannin(d-h) = O. The vertical eigenfunctions Zm(Z) and Qm(z) are defined as: ~cosh koz / sinh kod
Z,,(z)
m =0
(9a)
/
[cos k,,z/sinh kod
m >- 1
and
fcosh ~o(z-h)/sinh ~o(d-h)
Qm(z) = ~[cos ~m(z-h)/sinh ~o(d-h)
m =0 m >- 1 .
(9b)
In Equations (7a) and (7b), Ao and Bo represent the complex reflection and transmission coefficients which will be denoted by R and T, respectively. In order to obtain R and T, Equation (7a) is differentiated with respect to x and x is set to 0, and then both sides are multiplied by Zq(z), q = 0,1,2,..., and integrated over [0,d]. Utilizing the fact that Zm(Z) and Zq(z), m, q = 0,1,2 .... are orthogonals between [0, d], it yields:
Aq
1{
~qNq Ix°N°~°q +
f:
~)lx (O,z)
Zq(Z) dz
}
q = 0,1,...
(10)
where (d Nq = Jo ~q(z) dz
q = 0,1,....
(11)
Applying the matching condition (5a) and the structural condition (5c), and carrying out the integration, Equation (10) becomes
Aq = ~oq W ~qNq p~..o = ( Dpe-XpW - Cp) ~.pTpq,
q = 0,1,...
(12)
in which ~oq is the Kronecker delta, and the term Tpq is defined as
Tpq = f f Qp(z) Zq(z) dz,
p,q = 0,1,....
(13)
At the same interface, x = 0, Equation (7c) is written as 0¢
(I)3(O,z) :
Z
m=O
(Cm "q- Om e-xmW) Qm(Z).
(14)
688
A . G . ABuL-AzM
We then multiply both sides of Equation (14) by Qp(z) and integrate between [0,h], noting the orthogonality between the eigenfunctions Qm(z) and Qp(z) between [0,h]. The matching condition (5b) is now applied and the integration is carried out. Equation (14) becomes: 1(
Cp + Dpe-Xpw = ~
Tpo +
~ ) m=ommTpm ,
p = 0,1 ....
(15)
in which the term Sp is defined as
Sp =
)
QZ(z) dz,
p -- 0,1,....
(16)
Explicit expressions for No, Tpq and Sp defined in Equations (11), (13) and (16), respectively, are given in the Appendix. At the second interface, x = W, the same procedure will be followed and the matching conditions and the structural condition in Equation (6) are utilized. This will give a relation between the unknown coefficients B,n, Cm and Din, m = 0,1,2 ..... which is written as:
nq
--
[.I.qNq
p~=O(Op
Cpe-hp W + Op
--
Cpe-~'pw) hpTpq,
1
= Spp m=oZ nm
Tpm,
q = 0,1 ....
p = 0,1,....
(17)
(18)
Equations (15) and (18) are solved to obtain an expression for Cp and Dp, p = 0,1,2 .... in terms of Am and Bin. Substituting back into Equations (12) and (17) yields
Z
m=0
(mm(~)mq + ~-mq) -- Bm qlmq) = ~oq -- f--oq,
q = 0,1 ....
(19a)
oc
Z
m=O
(nm (~mq "[- E.mq) -- A m ~tgmq) = ~tfoq,
q = 0,1,...
(19b)
in which
~mq ~"
aPmq
Z hp Tpq TpmFlp ,,=0
(20a)
Sp~oNq
~'~ )kp Tpq TpmF2p p=OZ~ SplXqNq
(20b)
and = / i ( t a n BOW)-' Flp ((tanh f3oW)-1
p-1
//(sin f3oW)-1 F2p = ((sinh f~oW) 1
p-1
p=O
(21a)
p=O .
(21b)
Wide submerged breakwaters
689
Equations (19a) and (19b) now constitute two infinite simultaneous equations for the potential coefficients A m and Bm, m = 0,1,2,.... These equations may be truncated after a finite number of terms and the coefficients obtained by a standard matrixsolving technique. The pressure P at any point in the fluid may be written as P = -p~bt, where p is the fluid density. Then the total hydrodynamic force in the x-direction per unit length of the breakwater is given by Fx = f~ ipto [~l(0,z) - ~z(W,Z)] dz.
(22a)
Similarly the force component per unit length of the breakwater in the positive zdirection is given by: F~ = fow - ipo~~3(x,h) dx.
(22b)
PLANE WAVE APPROXIMATION In this approximation, only the most progressive mode of all modes in each region is used to satisfy the matching condition. The wave numbers ko in regions (1) and (2) and 13oin region (3) will be considered. In each region the velocity potential is simplified to: ePl = Z o ( z ) {e iko~°~° + R e -ikox~°~°}
(23a)
% = Zo(z) { Te" o
(23b)
¢b3 = Qo(z) {Coe - x ~ + DoeXo(x-W)}.
(23c)
Applying the matching conditions and the structural boundary condition in Equations (5) and (6), expressions for the reflection and transmission coefficients are given as: 1-A 2 R - l+A2 + 2ia
(24a)
2ib T - l+A2+2ia
(24b)
where A2 -----a 2 + b 2 and A -
iho T~oo kocos0 NoSo
(25a)
A a - tan(ihoW)
(25b)
and A b - sin(iXoW)"
(25c)
690
A.G. ABUL-AZM
The coefficients Co and Do are obtained as
iToo Co = { ( l + R ) e x o w - T} 2sin(iXoW)
(26a)
iTo Do = { TeXoW - ( I + R ) } 2sin(iXooW)"
(26b)
In Equation (24), R and T are functions of the structural admittance parameter A which depends on the submergence ratio h/d and 0 and the parameters a and b, which represent the breakwater width and the angle of wave attack, respectively. As h ~ 0, the function A ~ 1 and R ~ 0 independently of W. As 0 ~ 90°, h ~ ~ and ]R] 1 and [7] ~ 0. Here it should be noted that this approximate solution satisfies approximately the energy conservation law. This is due to the negligence of the evanescent modes, which are essential to satisfy the matching conditions. LONG WAVE APPROXIMATION In the case of the shallow water limit of the plane wave approximation, no evanescent modes are needed to satisfy the matching condition. The horizontal velocity of the long waves may be safely assumed to be uniform over the water depth. For oblique waves and when koW "~ 1 and hoW ~ 1, the function A 2 ~ 1 could be neglected. The following expressions are obtained for R and T: 1
R - I+(2A/XoW)
(27a)
and
T=
(2A/XoW) I+(2A/XoW).
(27b)
Here it is seen that the conservation of energy is satisfied, that is, R + T = 1 and IRI2 + 17]2 = 1 (Twersky, 1962). NUMERICAL RESULTS AND DISCUSSION In this section, numerical results are presented to illustrate the effect of different wave and structural parameters on the wave reflection, wave transmission and the hydrodynamic loads on the breakwater. In obtaining the numerical results presented herein the infinite summations in the expressions of the potentials were truncated after N terms. The simultaneous system of Equations (19a) and (19b) was solved using a truncation parameter that satisfies the conservation of energy relation within an error of O(10-5). Generally, N = 25 terms was found to give satisfactory results for all the numerical examples considered. Dimensionless parameters used in this section are as follows: submergence ratio h/d, the wave length parameter kod, width ratio W/d, wave length to water depth ratio Lo/d and wave steepness H/Lo. Forces are nondimensionalized by F' = 9gdH/2. The magnitude of the transmission coefficient IT] is presented for different wave and structural parameters in Figs 2-6. Numerical results are compared to those obtained analytically by Jeffreys (1944) for shallow water, and the experimental results obtained
Wide submerged breakwaters
691
I.I
LO
L
0.9
0.8
0.7
I
0.6
)0,5
0.4
0.3
0.2
0.1
~ •
0.0
METHOD
JEFFRYS
1944
EXP. ( J O H N S O N I
0.0
PRESENT
0.1
I
0.2
I
0.3
etal. 1951) I
0.4
I
I
0.5
0.6
I
0.7
I
0.8
I
0.9
I I.O
I.I
h/d FI6.2.
Comparison of transmission coefficients obtained by Jeffreys' theory and experiments: kod W/d = 0 . 3 6 8 a n d H/Lo = 0 . 0 1 9 5 .
= 0.666,
by Johnson et al. (1951), Abdul Khader and Rai (1980) and Dattatri et al. (1978). The expression below gives the value of T as obtained by Jeffreys (1944): T = (1 + 0.25 [ ~ / ~ _ h - ~ / - ~ ] 2
sin2c} -~
(28a)
where
c = ~koW tanh kod ~ ~----~h.
(28b)
In Fig. 2, the numerical results of the present method for normal wave incidence agree reasonably well with the experimental data points of Johnson et al. (1951). In this case/Cod = 0.666, W/d -- 0.368 and the wave steepness H/L o = 0.0195. Here it should be noted that in the experiment the wave steepness, which was not included in the context of the present linearized theory, has a significant effect--particularly as h/d increases. The experiments also consider the wave transmission due to overtopping for h/d > 1. For the same wave steepness and water depth, the width of the breakwater has increased as for the data in Figs 3 and 4, at which W/d = 0.66 and 0.991,
692
A.G. ABUL-AZM
-)l"
I,I
t.O
0.9
0.8 O.7
0,6
\
m
\
0,5
~d 0,4
0.3
0.2 PRESENT
METHOD
______ JEFFREYS
0,1
.)(, 0.0 0
1944
EXP. RESULTS (JOHNSON e t 8,] 1951J
I
i
0.1
0,2
I
0.3
I
0.4
I
0.5
I
I
0.6
0.7
I
O.S
I
I
0.9
1.0
II
h/d
FIG. 3. Comparisonof transmissioncoefficientsobtained by Jeffreys' theory and experiments: kod = 0.666, W/d = 0.66 and H/Lo = 0.0195. respectively. A sharply tuned peak, in both the present model and in Jeffreys' results as described by Wiegel (1964), is noted to occur at h i d = 0.98 and 0.95 in Figs 3 and 4, respectively. Compared to Figs 3 and 4, Fig. 5 shows a fairly good agreement with the experimental results obtained for higher wave steepness H / L o = 0.0295, shallower water kod = 0.861 and W / d = 0.664. Figure 6(a) and (b) shows that the present model results are closer to the experimental results of Abdul Khader and Rai (1980) and Dattatri et al. (1978) than those of Jeffreys. Figures 7-9 show the magnitude and phase of the reflection coefficient obtained from the present model against kod, for different values of W / d , 0 = 0 and h i d = 0.25, 0.05 and 0.75, respectively. The transmission coefficient was only shown in Fig. 9. In that case for h i d = 0.75, the reflection increases due to the increase of the breakwater height, and the transmission is decreased accordingly. Results for W / d = 0.01 are compared with those obtained by Abul-Azm (1993) for a thin, submerged and impermeable breakwater, that is, W / d = 0, using a different method, to check the correctness of the present solution. Figures 7-9 show that for a constant value of h/d, the reflection magnitude increases as W / d increases. Resonance occurs at some wave frequencies at which IR[ = 0, that is, the wave does not feel the breakwater, and a corresponding 180° change of phase is noticeable.
Wide submerged breakwaters
693
t.2
I,L
A
A
A
I.O
0.9
0.8
\\J
0.7
0.6
O.S
0.4
0.5
0.2
PRESENT ------
0.1
J
0.0
1944
EXP.(JOHNSON
I O.O
METHOD
JEFFREYS
0.1
I 0.2
etal.1951)
I
I
I
i
O.S
0.4
O.S
0.6
I
I
0.7
O.S
i 0.9
I 1.0
I.I
h/d
FIG. 4. Comparison of transmission coefficientsobtained by Jeffreys' theory and experiments: kod = 0.666, W/d = 0.991 and H/Lo = 0.0195. Figure 10 shows the magnitude and phase of the horizontal and vertical forces per unit length of the breakwater. As kod--> O, the nondimensional vertical force approaches the value of W/d. The effect of the wave angle 0 on IR[ and ITI is shown in Fig. 11, for different values of kod, W / d and h/d -- 0.25, 0.5 and 0.75. At some values of 0, IR[ will be equal to zero, with a corresponding increase in IT[ = 1, and as the wave becomes parallel to the breakwater, the reflection [R I becomes unity. This has been shown previously by many investigators (see, for example, Dalrymple et al., 1991). Figures 12 and 13 show the reflection coefficient as a function of kod, for some chosen values of 0, h/d and W/d. Of note is the general increase in IRI as 0 increases and the shift in the resonance point on the kod axis as the wave angle changes. The expression derived for the plane-wave approximation will now be discussed to aid the understanding of the reasons beyond the occurrence of the resonance points and the corresponding sharp peaks of the transmission. Equation (24a) shows that the reflection will equal zero in one of two independent conditions: IAI equals 1.0, which represents the admittance coefficient of the structure, or when the angle termed (ikoW) -- n~, n -- 1,2 .... , which depends on the structure width and the angle of wave attack. In the case of the full solution, for the same water depth and wave angle, the resonance
694
A.G.
ABUL-AZM
I.I t.O 0.9 0.8
\
0.7 0.6
\
0.5
0.4 0.3 0.2 ----
PRESENT
METHOD
JEFFREYS
1944
OA
"R"
EXP. RESULTS (JOHNSON e t
0.0 0.0
al. 1951)
I
I
I
I
I
I
I
I
i
O. I
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
i 1.0
II
h/d FIG. 5. C o m p a r i s o n of t r a n s m i s s i o n coefficients o b t a i n e d b y Jeffreys' t h e o r y a n d e x p e r i m e n t s : kod = 0.861, W/d = 0.664 a n d H/Lo = 0.0925.
1.2
1.2
(b)
(a)
~"
1.0
1.0
0.8
O.S g
0.6
0.4 0.2
0.6 0.4
PRESENT METHOD _____ JEFFREYS
•
ExR (o~r'rARi
er
0.2
AL)
A
JkI EXR i(A.KHp,OER p RAi),
0.0 0.0
O,Z
0"4
Ore
h/d
ole
,mO
0.0 1.2
PRESENT METHOD JEFFREYS
~----
EXP ( A . KHADER & RAI ) I I i I 0.2 Q4 ~6 0.8 hO I
O0
1.2
h/d
Fro. 6. C o m p a r i s o n of t r a n s m i s s i o n coefficients o b t a i n e d by Jeffreys' t h e o r y a n d e x p e r i m e n t s : (a) kod = 0.649, W/d = 0.473 a n d H/Lo = 0.0256; ( b ) kod = 1.095, W/d = 0.302 a n d H/Lo = 0.057.
will change according to how the structure will scatter waves, that is, the number of evanescent modes considered in the solution or the height and the width of the breakwater. Figures 14 and 15 show the results obtained by the two approximate solutions compared to the full solution results. For deep water, that is, small values of Lo/d, the plane-wave approximation gives a good estimate for the full solution results. This is particularly seen when the structure is small compared to the wave, which might be
Wide submerged breakwaters
695
0.20" (a) o.15.
W/d W/d W/d W/d
"......... --'~--
= = = =
0.5 1.0 0.0 0.01
f'. m
=__
0.10"
0
,
f 0.05"
"
~
.,,, 0.00"
1.0
0.0
2.0
3.0
4.0
5.0
6.0
kod
7.85 t (b)
"0 e~ L.
6.28 -
s
so
4'
/
:
: ,
...t
~
I t
1 ~-
!
~
. o,r /
:i
/|
:
•
s
S
.1
,,
l
" ,
! /
!
,w
1
i
'
4.71
3.14 0,0
1.0
2.0
3.0
4.0
5.0
6.0
kod F n 6 . 7 . ( a ) M a g n i t u d e a n d ( b ) p h a s e of the reflection coefficient as a f u n c t i o n of k o d (for d i f f e r e n t v a l u e s of W / d , h / d = 0.25 and 0 = 0).
696
A.G. ABUL-AZM 0.4
.....
0.3 t
- ....
W/d
=
0.5
W/d = 1.0
(a) ~,.,
W/d = 0.0
0.0 ~ 0.0
1.0
2.0
3.0
4.0
5.0
6.0
5.0
6.0
kod
7.85"
(b) / ! 8.28-
..';
•"
;."/f" ;
==
,,$ a
i /
l
,J
!
,,.
,
;
a
,..
,-:
!
,
I
at ,,,t=
3,14
0.0
1.0
2.0
3,0
~ 4.0
kod FIG. 8. (a) Magnitude and (b) phase of the reflection coefficient as a function of kod (for different values of W/d, h/d = 0.5 and 0 = 0).
Wide submerged breakwaters
697
1.2
I
(=) ....
1.0'
- .....................
.
I
0.8'
.......
W/d = 0.5
..........
W / d = 1.0
I
W / d = 0.0
0.6 ¸
,=,L
0.4' m
=_ 0.2'
0.0 0.0
1.0
2.0
3.0
4.0
5.0
6.0
k:
9.42
(b) a 6.28' .
L . _ . - - - - - - ~
.
.
.
.
.
.
.
.,,.,.,,
.
.
.
.
.
.
.
.
-~'-
3.14'
eL
-6.28
~ : 0.0
......... 1.0
; ......... 2.0
:.... 3.0
"."
..... . ......... 4.0
5.0
. 6.0
kod FiG. 9. ( a ) M a g n i t u d e and ( b ) phase o f t h e r e f l e c t i o n a n d t r a n s m i s s i o n c o e f f i c i e n t s as a f u n c t i o n o f d i f f e r e n t values o f W/d, h/d = 0.75 and 0 = 0 ) .
kod ( f o r
"
"g"
0
b
b
b
~
o
I
i
•
Phase of Fx (rad.)
m
o~
~
b
o
I~t
}~}
]Fx / F' I
N
>
>
>
G~
Wide submerged breakwaters
1.4
699
(c)
1.2 1.0 0.8
t~
0.6
0.2 0.41'0.0 0.0
110
210
310
410
-- ,I, 510 6.0
kod
7.85
(d)
6.28.
4.71 N
3.14 .,=
1
.
5
7
1.0
2.0
'
~
0.~' - I .57 '
0.0
3.0
4.0
5.0
6.0
kod FIG. 10. (a) Magnitude, (b) phase of horizontal force, (c) magnitude and (d) phase of vertical force as a function of kod (for different values of W / d and h/d, 0 = 0).
700
A. G. ABuL-AzM 1.2
(a)
1.o 0.8-
;,
tt
0.60.4-
ii I
X
$ 4 *1
=_
0.20.0
--
_
i
•
_ l
•
20
lO
-
_.
|
_.
•
30
-
4'0
•
,'0
|
8'0
¢0
80
90
T H E T A (Deg.)
1.2" (b) ~"
1.0-
--
0.8"
-
=__
-
kod=0•5, W]d --0.5
0.6"
--÷--
k/l=0.5, W/d=l.O
0.4"
I ........
krd=l.5, W/d=l.O
0
10
20
30
40
50
/~- /
,
' " . - ~
0.0
,,( ./~/~,:~I. [/////\
"-'--4--~-.4."
0.2
"',\'~,)
k~=~.5, w/d=o•5
-
,
60
70
,
80
90
T H E T A (Deg.)
1.2-
(c) _
1 .o-
.117.-. . . . . . . . . . .~. . . . . ~
~
]
~
~
0.8- " - ' P " P - ' P -'P
0"6• ~e
+" -,o- 4.._+. ,4,. ~
,,,'/ /'~,
..."//A"
0.4.
-
0.2, ,.
0.0
.
.
.
.
-°
.'" 10
20
30
40
50
60
70
80
go
T H E T A (Deg.) FI~• 11. Magnitude of reflection and transmission coefficients (a) h/d = 0.25, (b) hid = 0.50 and (c) hid = 0.75, as a function of 0 (for different values of/cod and W/d).
Wide submerged breakwaters 0.8
(,.)
0.6
6 = 15 ° -----o--- 9 = 3 0 " .... t .... 8 = 45 °
¸
..O II"
"~0.
"lD%
a •" 0.4
701
~.,
!!
---z,--.
O = 60 °
.... Q " "
8=75"
.....
0 = O*
oo
•
.
0.2, .m !_..'-
• ,,~ 0.0,
~ ..':llm~
~ -~ =,'-~, %
"u.
12.-" ~ ~ - ~ : ~ " -
•
0.0
,
•
-
1.0
" •
=.=.
,
,
~
2.0
3.0
4.0
-
--
5.0
6.0
kod
785] (b)
!
-
I
tA !
4.71 "~'
i
! ~
,d~r'.,..O ' ~ r =_..ao.a°"~ '~,
]
,.i,~!l
s,, J 1.57 1 0.~!
0.0
!
110
210
:310
4.0!
5.0
6.0
koa P[o. 12. (a) Magnitude and (b) phase of the reflection coefficient as a function of/cod (for W / d = 1.0 and variable wave angles).
hid
= 0.5,
702
A . G . ABUL-AZM 0.8
(a) o,D
0.6 ¸
o e ' 4 " 41' ° 4 ° "° ' m " "° " '1' • "° ° ~
i,
;
"e..,e
!
m
"e.
0.4' ~-o-O-,I~I-o. a ;P 0.2"
!A
,
"..
I/,;.
~a~D~.ID
~
"u.
V~\
.-"""""~',~.
0.0 0,0
1.0
2.0
3.0
4.0
5.0
6.0
kod
I
j (b)
785 I
~I ~ =¢
'"t
eL
3.14
I
,11,,41
0.~0°A°
1.57
t"
0.00 / 0.0
i
!
i
!
!
1.0
2.0
3.0
4.0
5.0
6.0
kod FIG. 13. (a) Magnitude and (b) phase of the reflection coefficient as a function of kod (for h/d = 0.75, W/d = 0.5 and different wave angles); for legend, see Fig. 12.
Wide submerged breakwaters
703
1.o
(a) Full Solution Plane-Wave Approx. Long Wave Approx.
0.8 % % \
0.6" m
0.4"
% %% ,%
0.2'
0.0
,% %,
10
100
Lo/d
9.42 "t
(b)
7.85 "4
\
¢1
m
6.28
¢1
,,= 4.71
3
I
.
1
4
10
~
100
Lo/d FIG. 14. Comparison of (a) magnitude and (b) phase of the reflection coefficient as a function of Lo/d, using different theories (for h/d = 0.25, w/d = 0.5 and 0 = 0).
704
A.G.
AauL-AzM
1.0. •~
(a)
~,
" ... ~-
Full Solution - - -
0.8-
. ...........
.....
P l a n e - W a v e Approx. L o n g W a v e Approx.
\4
0.6
0.4
0.2
0.0 10
100
LJd
9,42 (b)
7.85 I
•=
i
6.28
1
¢I
4.71 1
"¢
3.14
1.57
0.~
. . . . . . . .
1'0
. . . . . . .
I00
Lo/d FIG. 15. Comparison o f (a) magnitude and (b) phase o f reflection coefficient as a function o f Lo/d, using different theories ( f o r hid = 0.75, w/d = 1.0 and 0 = 0).
Wide submerged breakwaters
705
e x p l a i n e d in t e r m s o f less i n t e r f e r e n c e with t h e free surface, as s h o w n in Fig. 14. A s t h e w a v e l e n g t h L o / d i n c r e a s e s , t h e l o n g - w a v e a p p r o x i m a t e results t e n d to h a v e a b e t t e r r e p r e s e n t a t i o n o f t h e full s o l u t i o n results, as s h o w n in Fig. 15. CONCLUSION A n e x a c t m e t h o d b a s e d o n e i g e n f u n c t i o n e x p a n s i o n t e c h n i q u e has b e e n utilized to d e t e r m i n e t h e e f f e c t i v e n e s s o f a w i d e , s u b m e r g e d a n d i m p e r m e a b l e b r e a k w a t e r in reflecting o b l i q u e waves. H y d r o d y n a m i c h o r i z o n t a l a n d v e r t i c a l forces h a v e b e e n c o m p u t e d o n t h e b r e a k w a t e r . U n d e r t h e a s s u m p t i o n o f l i n e a r w a v e s a n d p o t e n t i a l flow t h e o r i e s , t h e n u m e r i c a l results o b t a i n e d f r o m t h e p r e s e n t m o d e l h a v e s h o w n a g o o d a g r e e m e n t with t h e c o r r e s p o n d i n g e x p e r i m e n t a l results o b t a i n e d b y o t h e r i n v e s t i g a t o r s , p a r t i c u l a r l y for flatter waves. T h e t h e o r y has b e e n e x t e n d e d to o b t a i n t w o a p p r o x i m a t e solutions, n a m e l y t h e p l a n e - w a v e a p p r o x i m a t i o n a n d t h e l o n g - w a v e solution. T h e results o b t a i n e d f r o m t h e two a p p r o x i m a t i o n s h a v e b e e n s h o w n to give a g o o d e s t i m a t e for the full s o l u t i o n results in d e e p a n d shallow w a t e r , r e s p e c t i v e l y .
REFERENCES ABuL-AzM, A.G. 1993. Wave diffraction through submerged breakwaters. J. WatWay, Port, Coastal Ocean Engng, ASCE 119, 587-605. ABDUL KHADER,M.H. and RAL S.P. 1980. A study of submerged breakwaters. J. Hydr. Res. 18, 113-121. DALRYMPLE,R.A., LOSADA,M.A. and MASON,P.A. 1991. Reflection and transmission from porous structures under oblique wave attack. J. Fluid Mech. 224, 625-644. DALRYMPLE,R.A. and MAi~TIr~,P.A. 1990. Wave diffraction through offshore breakwaters. J. WatWay, Port, Coastal Ocean Engng, ASCE 116, 727-741. DAITATRI,J., RA~Ar~, H. and SnANKAR,J. 1978. Performance characteristics of submerged breakwaters. Proceedings of the 16th Coastal Engineering Conference, ASCE, Hamburg. DEAN, W.R. 1945. On the reflection of the surface waves by submerged plane barrier. Proc. Cambridge Phil. Soc. 4, 231-236. DICK, T.M. and BREBNER,A. 1968. Solid and permeable submerged breakwaters. In Proceedings of the llth Coastal Engineering Conference, ASCE, London, Vol. II, pp. 1141-1158. HALL,W.C. and HALL,J.V. 1940. A model study of the effect of submerged breakwaters on wave action. U.S. Army Corps of Engineers, Beach Erosion Board, Technical Memorandum, No. 1, May 1940. JEFFREYS,H. 1944. Note on the offshore bar problems and reflection from a bar. Great Britain Ministry of Supply, Wave Report No. 3, June 1944. JOHr~SON, J.W., Fucns, R.A. and Movasor~, J.R. 1951. The damping action of submerged breakwaters. Trans. Amer. Geophys. Union 32, 704-718. KOBAYASltl,N. and WORTANJO,A. 1989. Wave transmission over submerged breakwaters. J. WatWay, Port, Coastal Ocean Engng, ASCE 115, 662-680. MADSEN, P.A. 1983. Wave reflection from a vertical permeable wave absorber. Coastal Engng 7, 381-396. SABOr~C~,T. and GORES, O. 1992. Wave propagation over sloping bottom and submerged breakwaters. In Proceedings of the 2nd International Marina Technical Conference, Southampton, U.K., pp. 185-195. SAV.Pr,AVA, T. and ISAACSON,M. 1981. Mechanics of Wave Forces on Offshore Structures. Van NostrandReinhold, New York. SILVESrER,R. 1976. Headlands defence of coasts. In Proceedings of the 15th Coastal Engineering Conference, pp. 1394-1406. SOLLITF,C.K. and CROSS, R.H. 1972. Wave reflection and transmission at permeable breakwaters. Department of Civil Engineering, School of Engineering, MIT, Report No. 147, March 1972. SuLisz, W. 1985. Wave reflection and transmission at permeable breakwaters of arbitrary cross-section. Coastal Engng 9, 371-386. TWERSKV, V. 1962. On the scattering of waves by the infinite grating of circular cylinders. IRE Trans. Antennas Propag. 10, 737-765. WIEGEL,R.L. 1964. Oceanographical engineering. In Theoretical and Applied Fluid Mechanics. PrenticeHall, Englewood Cliffs.
706
A . O . ABut:Azr~
APPENDIX In this Appendix, explicit expressions are given for the z-integrals appearing in Equations (11), (13) and (16): sinh 2kod + 2kod
4ko sinh2kod q=0 Nq = sin 2kqd + 2kqd q >- 1 4kq sinh2kod [- sinh 26o(d-h ) + 2f3o(d-h ) Sp = [ sin 213p(d-h) + 2~e(d-h) t - - 4~3ps i n h 2 ~ ( d - h ) sinh
Too =
(A1)
(A2) p -- 1
(koz+fgo(z-h)) sinh (koz-~o(z-h)) i" O-1 2(ko+fSo) ~2(ko-~o)
Toq = {f3ocoskqzsinh~3o(z-h)(f32o + k~) + kqsinkqzcosh~3o(z-h)(d_h) --
(A3)
i • D -l ,
q -> 1
(A4)
Tpo = {f3ocoshkozsinf3p(z-h)+(f3zp + kz)k°sinhk°zc°sf3e(z-h)(d-h) i • D - ' ,
p -> 1
(A5)
Isinkqz+f3p(z+h)) sin(kqz-f3p(z-h)) i Tpq [ 2(kq-+~p) + 2(kq'~t,~ " D-I' =
and D = sinh
kod sinh fSo(d-h).
P'
q-> 1
(A6)
~)
Pergamon
Copyright © 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 00"29-8018194 $7.00 + .00
DIFFRACTION THROUGH BREAKWATERS UNDER
WIDE SUBMERGED OBLIQUE WAVES
A. G. ABUL-AZM Irrigation and Hydraulics Department, Faculty of Engineering, Cairo University, Giza, Egypt Abstract--The eigenfunction expansion method is used to develop a theoretical solution for the flow field caused by monochromatic oblique waves incident upon an infinitely long, submerged, wide and vertical breakwater situated in arbitrary water depth. The breakwater is assumed to be uniform in the y-direction, rigid and impermeable, and the fluid motion is idealized as linearized, two-dimensional potential flow. Numerical results are presented to show the effect of different wave and structural parameters on the transmitted and reflected waves and the hydrodynamic loadings on the breakwater. The accuracy of the solution is verified by comparing the numerical results with the available theoretical or experimental results. Two different simplified solutions for the problem are presented under the approximation of planewaves and under the assumption of long oblique waves.
INTRODUCTION
THE MAIN function of a breakwater is to protect the shoreward area in the leeside of the structure by reflecting or breaking the incident waves. Offshore, construction of a conventional type breakwater--which extends above the highest expected sea level-may have some drawbacks, such as the high construction cost and the obstruction of the view, particularly in front of a pleasure beach or a marina. A wide submerged breakwater may be used to overcome these advantages, and if properly designed, it may significantly reduce the wave energy near shore. Recently, submerged breakwaters have been constructed in prone areas to replace coral reefs in controlling beach erosion or as permanent headlands for the formation of crenulate beaches; see Silvester (1976). Wave propagation over various two-dimensional underwater structures has been widely studied. SoUitt and Cross (1972) and Madsen (1983) have considered the dissipation of wave energy inside rectangular, emerged and porous structures under normal wave incidence. Sulisz (1985) solved the problem for arbitrary cross-section and Dalrymple et al. (1991) have utilized the eigenfunction method to show that for oblique waves incident upon a vertical porous structure, the reflection and transmission coefficients are significantly altered. Wave propagation over sloping and submerged breakwaters has also been studied analytically by Kobayashi and Wurtanjo (1989) and Sabuncu and Goren (1992), who presented an approximate method based on representing the smooth slope by a stepwise geometry and matching the boundary conditions at each step. The problem of a rigid, thin, submerged and impermeable barrier, which does not take into consideration the width effect, has been studied theoretically by many investigators, such as Dean (1945) in the case of infinitely deep water, and Johnson 683
684
A, G. ABuL-AZM
et al. (1951), who provided an approximate solution based on equating the incident
and the transmitted wave powers, and presented experimental results for a submerged rectangular breakwater. Recently, Abul-Azm (1993) presented a solution to determine the flow field around different types of thin, rigid and submerged breakwaters. The method used was the eigenfunction technique together with the least-squares method for the solution of the mixed boundary condition. This method is proven to be analytically straightforward and numerically efficient, as shown by Dalrymple and Martin (1990) in predicting the wave reflection for a long linear array of offshore detached breakwaters. The effectiveness of rigid, wide and submerged breakwaters to dissipate wave energy had been studied experimentally by Hall and Hall (1940), Dick and Brebner (1968), Dattatri et al. (1978) and Abdul Khader and Rai (1980). Jeffreys (1944) had presented an approximate analytical expression to compute the wave transmission through wide submerged breakwaters situated in shallow waters. In this paper the eigenfunction expansion method is adopted to estimate the flow potential around a rigid, impermeable, wide rectangular and submerged breakwater subjected to oblique waves. Numerical results for the transmission coefficients are compared with the experimental results of Jeffreys (1944), Dattatri et al. (1978) and Abdul Khader and Rai (1980). Further numerical results are presented to show the effect of different wave and structural parameters on the transmission coefficient, reflection coefficient and the hydrodynamic horizontal and vertical forces on the breakwater. Simplified analytical solutions are presented using the plane-wave approximation and the long wave assumption. Numerical results obtained from the approximation methods are used to illustrate the validity of the solutions. THEORETICAL FORMULATION The geometry of the submerged breakwater under investigation is depicted in Fig. 1. The system is idealized as two-dimensional, and Cartesian coordinates are employed. The breakwater is assumed to be rigid and impermeable, and has a width W and height h above the sea-bed. The z-axis is directed vertically upwards from an origin on the sea-bed in water of arbitrary depth d. The fluid domain is divided into three regions: region (1) for x - 0, region 2 for x --- W and region (3) for 0 < x < W. The breakwater is subjected to an incident train of regular, monochromatic, small amplitude waves of height H and frequency (o, obliquely propagating with an angle 0 measured anticlockwise from the positive x-direction as shown in Fig. 1. The fluid is assumed to be inviscid, incompressible and the motion is irrotational. The velocity potential ( x , y , z , t ) can be expressed as (oH
dpi(x'Y'Z't)Re = ~ o f~i(X'z)ei(k°ysinO--t°t)'
i = 1,2,3
(1)
where Re is the real part of the complex expression and ko is the wave number which is related to the angular frequency through the dispersion relation (o2 = gko tanhkod. Here it should be noted that the y-variation of the solution must be the same in each region (Snell's Law)--see Dalrymple et al. (1991)--in order to later satisfy the matching condition at each vertical interface. The fluid velocity vector is given by qi = V~i; then the following boundary-valueproblem is obtained in each region i = 1,2,3 (Sarpkaya and Isaacson, 1981) as follows:
685
Wide submerged breakwaters at
HI v
HT
HR
r
(1)
(3)
(2)
w
m
X
FIG. 1. Definition sketch.
V2(~i = 0
Oz
- 0
~(I) i
g~
-
o)2(I)i
=
0
everywhere in the region of flow
(2a)
o n z = 0 f o r i = 1,2 and z = h f o r i = 3
(2b)
on z = d
(2c)
where g is the acceleration due to gravity. In subsequent sections, the total velocity potential will be decomposed into incident and scattered potentials, namely ~ = ~z + ~s. At large distances from the breakwater, • s must satisfy a radiation condition, namely
686
A . G . ABuL-AZM
lim ( O - T - i k o ] ( ~ i s ) = O ,
\ox
i= land2
/
where the incident potential
(I) I
(3)
is given by
* , = Zo(z) e ikoc°s°
(4)
and Zo(z) is the depth dependency function. The solution across each interface, that is, x = 0 and x = W, must be continuous and satisfy the continuity of mass flux and pressure; thus it requires that at x = 0: ( I ) l x = (I)3x ,
h -< z -< d
(5a)
~1 = qb3,
h-< z -
(5b)
d
and from the structural boundary conditions:
(5c)
O<-z<-h.
~lx = 0, Similarly at x = W (I)2x = (I)3x ,
h -< z -< d
(6a)
(I~ 2 = (I)3,
h < z -< d
(6b)
and (6c)
O<_z<--h
qb2~=0 ,
where the subscript indicates derivative with respect to x. EIGENFUNCTION EXPANSION SOLUTION An appropriate solution for the potentials which satisfies Equation (2) may be written as: oe
fl)l = Zo(Z ) ei,%~coso + ~
A,,, e ~,,'~ Zm(z)
(7a)
m=O
fl)2 = ~
n m e -~,,,(x-w) Z m ( Z )
(7b)
{Cm e-xmx + D,.,, ex'(x-w)} Qm(z).
(7c)
m=O oo
~3 = ~ m=O
Am, Bin, Cm and Dm, m = 0,1,2,... represent the unknown potential coefficients which are yet to be determined through the interface conditions (5) and (6). In the summation terms:
-ikocosO
~
and
I = (~k~
m= 0 (8a)
+ k~sin20
m ~ 1
Wide submerged breakwaters
•m =
- i X/132- k2sin20
m = 0
,/ 132 + k2sin20
m -> 1
687
(8b)
where km, m -> 1 are the evanescent wave modes in regions (1) and (2), which are determined as the positive real roots of the transcendental equation, oa2+gkm tankmd = O. 13o is the wave number inside region (3), computed as the real root of the dispersion relation (DE -- g~o tanh~o(d-h) = 0 and [~,,, m -> 1 are its associated evanescent modes obtained from the expression (D2+g[3,,, tannin(d-h) = O. The vertical eigenfunctions Zm(Z) and Qm(z) are defined as: ~cosh koz / sinh kod
Z,,(z)
m =0
(9a)
/
[cos k,,z/sinh kod
m >- 1
and
fcosh ~o(z-h)/sinh ~o(d-h)
Qm(z) = ~[cos ~m(z-h)/sinh ~o(d-h)
m =0 m >- 1 .
(9b)
In Equations (7a) and (7b), Ao and Bo represent the complex reflection and transmission coefficients which will be denoted by R and T, respectively. In order to obtain R and T, Equation (7a) is differentiated with respect to x and x is set to 0, and then both sides are multiplied by Zq(z), q = 0,1,2,..., and integrated over [0,d]. Utilizing the fact that Zm(Z) and Zq(z), m, q = 0,1,2 .... are orthogonals between [0, d], it yields:
Aq
1{
~qNq Ix°N°~°q +
f:
~)lx (O,z)
Zq(Z) dz
}
q = 0,1,...
(10)
where (d Nq = Jo ~q(z) dz
q = 0,1,....
(11)
Applying the matching condition (5a) and the structural condition (5c), and carrying out the integration, Equation (10) becomes
Aq = ~oq W ~qNq p~..o = ( Dpe-XpW - Cp) ~.pTpq,
q = 0,1,...
(12)
in which ~oq is the Kronecker delta, and the term Tpq is defined as
Tpq = f f Qp(z) Zq(z) dz,
p,q = 0,1,....
(13)
At the same interface, x = 0, Equation (7c) is written as 0¢
(I)3(O,z) :
Z
m=O
(Cm "q- Om e-xmW) Qm(Z).
(14)
688
A . G . ABuL-AzM
We then multiply both sides of Equation (14) by Qp(z) and integrate between [0,h], noting the orthogonality between the eigenfunctions Qm(z) and Qp(z) between [0,h]. The matching condition (5b) is now applied and the integration is carried out. Equation (14) becomes: 1(
Cp + Dpe-Xpw = ~
Tpo +
~ ) m=ommTpm ,
p = 0,1 ....
(15)
in which the term Sp is defined as
Sp =
)
QZ(z) dz,
p -- 0,1,....
(16)
Explicit expressions for No, Tpq and Sp defined in Equations (11), (13) and (16), respectively, are given in the Appendix. At the second interface, x = W, the same procedure will be followed and the matching conditions and the structural condition in Equation (6) are utilized. This will give a relation between the unknown coefficients B,n, Cm and Din, m = 0,1,2 ..... which is written as:
nq
--
[.I.qNq
p~=O(Op
Cpe-hp W + Op
--
Cpe-~'pw) hpTpq,
1
= Spp m=oZ nm
Tpm,
q = 0,1 ....
p = 0,1,....
(17)
(18)
Equations (15) and (18) are solved to obtain an expression for Cp and Dp, p = 0,1,2 .... in terms of Am and Bin. Substituting back into Equations (12) and (17) yields
Z
m=0
(mm(~)mq + ~-mq) -- Bm qlmq) = ~oq -- f--oq,
q = 0,1 ....
(19a)
oc
Z
m=O
(nm (~mq "[- E.mq) -- A m ~tgmq) = ~tfoq,
q = 0,1,...
(19b)
in which
~mq ~"
aPmq
Z hp Tpq TpmFlp ,,=0
(20a)
Sp~oNq
~'~ )kp Tpq TpmF2p p=OZ~ SplXqNq
(20b)
and = / i ( t a n BOW)-' Flp ((tanh f3oW)-1
p-1
//(sin f3oW)-1 F2p = ((sinh f~oW) 1
p-1
p=O
(21a)
p=O .
(21b)
Wide submerged breakwaters
689
Equations (19a) and (19b) now constitute two infinite simultaneous equations for the potential coefficients A m and Bm, m = 0,1,2,.... These equations may be truncated after a finite number of terms and the coefficients obtained by a standard matrixsolving technique. The pressure P at any point in the fluid may be written as P = -p~bt, where p is the fluid density. Then the total hydrodynamic force in the x-direction per unit length of the breakwater is given by Fx = f~ ipto [~l(0,z) - ~z(W,Z)] dz.
(22a)
Similarly the force component per unit length of the breakwater in the positive zdirection is given by: F~ = fow - ipo~~3(x,h) dx.
(22b)
PLANE WAVE APPROXIMATION In this approximation, only the most progressive mode of all modes in each region is used to satisfy the matching condition. The wave numbers ko in regions (1) and (2) and 13oin region (3) will be considered. In each region the velocity potential is simplified to: ePl = Z o ( z ) {e iko~°~° + R e -ikox~°~°}
(23a)
% = Zo(z) { Te" o
(23b)
¢b3 = Qo(z) {Coe - x ~ + DoeXo(x-W)}.
(23c)
Applying the matching conditions and the structural boundary condition in Equations (5) and (6), expressions for the reflection and transmission coefficients are given as: 1-A 2 R - l+A2 + 2ia
(24a)
2ib T - l+A2+2ia
(24b)
where A2 -----a 2 + b 2 and A -
iho T~oo kocos0 NoSo
(25a)
A a - tan(ihoW)
(25b)
and A b - sin(iXoW)"
(25c)
690
A.G. ABUL-AZM
The coefficients Co and Do are obtained as
iToo Co = { ( l + R ) e x o w - T} 2sin(iXoW)
(26a)
iTo Do = { TeXoW - ( I + R ) } 2sin(iXooW)"
(26b)
In Equation (24), R and T are functions of the structural admittance parameter A which depends on the submergence ratio h/d and 0 and the parameters a and b, which represent the breakwater width and the angle of wave attack, respectively. As h ~ 0, the function A ~ 1 and R ~ 0 independently of W. As 0 ~ 90°, h ~ ~ and ]R] 1 and [7] ~ 0. Here it should be noted that this approximate solution satisfies approximately the energy conservation law. This is due to the negligence of the evanescent modes, which are essential to satisfy the matching conditions. LONG WAVE APPROXIMATION In the case of the shallow water limit of the plane wave approximation, no evanescent modes are needed to satisfy the matching condition. The horizontal velocity of the long waves may be safely assumed to be uniform over the water depth. For oblique waves and when koW "~ 1 and hoW ~ 1, the function A 2 ~ 1 could be neglected. The following expressions are obtained for R and T: 1
R - I+(2A/XoW)
(27a)
and
T=
(2A/XoW) I+(2A/XoW).
(27b)
Here it is seen that the conservation of energy is satisfied, that is, R + T = 1 and IRI2 + 17]2 = 1 (Twersky, 1962). NUMERICAL RESULTS AND DISCUSSION In this section, numerical results are presented to illustrate the effect of different wave and structural parameters on the wave reflection, wave transmission and the hydrodynamic loads on the breakwater. In obtaining the numerical results presented herein the infinite summations in the expressions of the potentials were truncated after N terms. The simultaneous system of Equations (19a) and (19b) was solved using a truncation parameter that satisfies the conservation of energy relation within an error of O(10-5). Generally, N = 25 terms was found to give satisfactory results for all the numerical examples considered. Dimensionless parameters used in this section are as follows: submergence ratio h/d, the wave length parameter kod, width ratio W/d, wave length to water depth ratio Lo/d and wave steepness H/Lo. Forces are nondimensionalized by F' = 9gdH/2. The magnitude of the transmission coefficient IT] is presented for different wave and structural parameters in Figs 2-6. Numerical results are compared to those obtained analytically by Jeffreys (1944) for shallow water, and the experimental results obtained
Wide submerged breakwaters
691
I.I
LO
L
0.9
0.8
0.7
I
0.6
)0,5
0.4
0.3
0.2
0.1
~ •
0.0
METHOD
JEFFRYS
1944
EXP. ( J O H N S O N I
0.0
PRESENT
0.1
I
0.2
I
0.3
etal. 1951) I
0.4
I
I
0.5
0.6
I
0.7
I
0.8
I
0.9
I I.O
I.I
h/d FI6.2.
Comparison of transmission coefficients obtained by Jeffreys' theory and experiments: kod W/d = 0 . 3 6 8 a n d H/Lo = 0 . 0 1 9 5 .
= 0.666,
by Johnson et al. (1951), Abdul Khader and Rai (1980) and Dattatri et al. (1978). The expression below gives the value of T as obtained by Jeffreys (1944): T = (1 + 0.25 [ ~ / ~ _ h - ~ / - ~ ] 2
sin2c} -~
(28a)
where
c = ~koW tanh kod ~ ~----~h.
(28b)
In Fig. 2, the numerical results of the present method for normal wave incidence agree reasonably well with the experimental data points of Johnson et al. (1951). In this case/Cod = 0.666, W/d -- 0.368 and the wave steepness H/L o = 0.0195. Here it should be noted that in the experiment the wave steepness, which was not included in the context of the present linearized theory, has a significant effect--particularly as h/d increases. The experiments also consider the wave transmission due to overtopping for h/d > 1. For the same wave steepness and water depth, the width of the breakwater has increased as for the data in Figs 3 and 4, at which W/d = 0.66 and 0.991,
692
A.G. ABUL-AZM
-)l"
I,I
t.O
0.9
0.8 O.7
0,6
\
m
\
0,5
~d 0,4
0.3
0.2 PRESENT
METHOD
______ JEFFREYS
0,1
.)(, 0.0 0
1944
EXP. RESULTS (JOHNSON e t 8,] 1951J
I
i
0.1
0,2
I
0.3
I
0.4
I
0.5
I
I
0.6
0.7
I
O.S
I
I
0.9
1.0
II
h/d
FIG. 3. Comparisonof transmissioncoefficientsobtained by Jeffreys' theory and experiments: kod = 0.666, W/d = 0.66 and H/Lo = 0.0195. respectively. A sharply tuned peak, in both the present model and in Jeffreys' results as described by Wiegel (1964), is noted to occur at h i d = 0.98 and 0.95 in Figs 3 and 4, respectively. Compared to Figs 3 and 4, Fig. 5 shows a fairly good agreement with the experimental results obtained for higher wave steepness H / L o = 0.0295, shallower water kod = 0.861 and W / d = 0.664. Figure 6(a) and (b) shows that the present model results are closer to the experimental results of Abdul Khader and Rai (1980) and Dattatri et al. (1978) than those of Jeffreys. Figures 7-9 show the magnitude and phase of the reflection coefficient obtained from the present model against kod, for different values of W / d , 0 = 0 and h i d = 0.25, 0.05 and 0.75, respectively. The transmission coefficient was only shown in Fig. 9. In that case for h i d = 0.75, the reflection increases due to the increase of the breakwater height, and the transmission is decreased accordingly. Results for W / d = 0.01 are compared with those obtained by Abul-Azm (1993) for a thin, submerged and impermeable breakwater, that is, W / d = 0, using a different method, to check the correctness of the present solution. Figures 7-9 show that for a constant value of h/d, the reflection magnitude increases as W / d increases. Resonance occurs at some wave frequencies at which IR[ = 0, that is, the wave does not feel the breakwater, and a corresponding 180° change of phase is noticeable.
Wide submerged breakwaters
693
t.2
I,L
A
A
A
I.O
0.9
0.8
\\J
0.7
0.6
O.S
0.4
0.5
0.2
PRESENT ------
0.1
J
0.0
1944
EXP.(JOHNSON
I O.O
METHOD
JEFFREYS
0.1
I 0.2
etal.1951)
I
I
I
i
O.S
0.4
O.S
0.6
I
I
0.7
O.S
i 0.9
I 1.0
I.I
h/d
FIG. 4. Comparison of transmission coefficientsobtained by Jeffreys' theory and experiments: kod = 0.666, W/d = 0.991 and H/Lo = 0.0195. Figure 10 shows the magnitude and phase of the horizontal and vertical forces per unit length of the breakwater. As kod--> O, the nondimensional vertical force approaches the value of W/d. The effect of the wave angle 0 on IR[ and ITI is shown in Fig. 11, for different values of kod, W / d and h/d -- 0.25, 0.5 and 0.75. At some values of 0, IR[ will be equal to zero, with a corresponding increase in IT[ = 1, and as the wave becomes parallel to the breakwater, the reflection [R I becomes unity. This has been shown previously by many investigators (see, for example, Dalrymple et al., 1991). Figures 12 and 13 show the reflection coefficient as a function of kod, for some chosen values of 0, h/d and W/d. Of note is the general increase in IRI as 0 increases and the shift in the resonance point on the kod axis as the wave angle changes. The expression derived for the plane-wave approximation will now be discussed to aid the understanding of the reasons beyond the occurrence of the resonance points and the corresponding sharp peaks of the transmission. Equation (24a) shows that the reflection will equal zero in one of two independent conditions: IAI equals 1.0, which represents the admittance coefficient of the structure, or when the angle termed (ikoW) -- n~, n -- 1,2 .... , which depends on the structure width and the angle of wave attack. In the case of the full solution, for the same water depth and wave angle, the resonance
694
A.G.
ABUL-AZM
I.I t.O 0.9 0.8
\
0.7 0.6
\
0.5
0.4 0.3 0.2 ----
PRESENT
METHOD
JEFFREYS
1944
OA
"R"
EXP. RESULTS (JOHNSON e t
0.0 0.0
al. 1951)
I
I
I
I
I
I
I
I
i
O. I
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
i 1.0
II
h/d FIG. 5. C o m p a r i s o n of t r a n s m i s s i o n coefficients o b t a i n e d b y Jeffreys' t h e o r y a n d e x p e r i m e n t s : kod = 0.861, W/d = 0.664 a n d H/Lo = 0.0925.
1.2
1.2
(b)
(a)
~"
1.0
1.0
0.8
O.S g
0.6
0.4 0.2
0.6 0.4
PRESENT METHOD _____ JEFFREYS
•
ExR (o~r'rARi
er
0.2
AL)
A
JkI EXR i(A.KHp,OER p RAi),
0.0 0.0
O,Z
0"4
Ore
h/d
ole
,mO
0.0 1.2
PRESENT METHOD JEFFREYS
~----
EXP ( A . KHADER & RAI ) I I i I 0.2 Q4 ~6 0.8 hO I
O0
1.2
h/d
Fro. 6. C o m p a r i s o n of t r a n s m i s s i o n coefficients o b t a i n e d by Jeffreys' t h e o r y a n d e x p e r i m e n t s : (a) kod = 0.649, W/d = 0.473 a n d H/Lo = 0.0256; ( b ) kod = 1.095, W/d = 0.302 a n d H/Lo = 0.057.
will change according to how the structure will scatter waves, that is, the number of evanescent modes considered in the solution or the height and the width of the breakwater. Figures 14 and 15 show the results obtained by the two approximate solutions compared to the full solution results. For deep water, that is, small values of Lo/d, the plane-wave approximation gives a good estimate for the full solution results. This is particularly seen when the structure is small compared to the wave, which might be
Wide submerged breakwaters
695
0.20" (a) o.15.
W/d W/d W/d W/d
"......... --'~--
= = = =
0.5 1.0 0.0 0.01
f'. m
=__
0.10"
0
,
f 0.05"
"
~
.,,, 0.00"
1.0
0.0
2.0
3.0
4.0
5.0
6.0
kod
7.85 t (b)
"0 e~ L.
6.28 -
s
so
4'
/
:
: ,
...t
~
I t
1 ~-
!
~
. o,r /
:i
/|
:
•
s
S
.1
,,
l
" ,
! /
!
,w
1
i
'
4.71
3.14 0,0
1.0
2.0
3.0
4.0
5.0
6.0
kod F n 6 . 7 . ( a ) M a g n i t u d e a n d ( b ) p h a s e of the reflection coefficient as a f u n c t i o n of k o d (for d i f f e r e n t v a l u e s of W / d , h / d = 0.25 and 0 = 0).
696
A.G. ABUL-AZM 0.4
.....
0.3 t
- ....
W/d
=
0.5
W/d = 1.0
(a) ~,.,
W/d = 0.0
0.0 ~ 0.0
1.0
2.0
3.0
4.0
5.0
6.0
5.0
6.0
kod
7.85"
(b) / ! 8.28-
..';
•"
;."/f" ;
==
,,$ a
i /
l
,J
!
,,.
,
;
a
,..
,-:
!
,
I
at ,,,t=
3,14
0.0
1.0
2.0
3,0
~ 4.0
kod FIG. 8. (a) Magnitude and (b) phase of the reflection coefficient as a function of kod (for different values of W/d, h/d = 0.5 and 0 = 0).
Wide submerged breakwaters
697
1.2
I
(=) ....
1.0'
- .....................
.
I
0.8'
.......
W/d = 0.5
..........
W / d = 1.0
I
W / d = 0.0
0.6 ¸
,=,L
0.4' m
=_ 0.2'
0.0 0.0
1.0
2.0
3.0
4.0
5.0
6.0
k:
9.42
(b) a 6.28' .
L . _ . - - - - - - ~
.
.
.
.
.
.
.
.,,.,.,,
.
.
.
.
.
.
.
.
-~'-
3.14'
eL
-6.28
~ : 0.0
......... 1.0
; ......... 2.0
:.... 3.0
"."
..... . ......... 4.0
5.0
. 6.0
kod FiG. 9. ( a ) M a g n i t u d e and ( b ) phase o f t h e r e f l e c t i o n a n d t r a n s m i s s i o n c o e f f i c i e n t s as a f u n c t i o n o f d i f f e r e n t values o f W/d, h/d = 0.75 and 0 = 0 ) .
kod ( f o r
"
"g"
0
b
b
b
~
o
I
i
•
Phase of Fx (rad.)
m
o~
~
b
o
I~t
}~}
]Fx / F' I
N
>
>
>
G~
Wide submerged breakwaters
1.4
699
(c)
1.2 1.0 0.8
t~
0.6
0.2 0.41'0.0 0.0
110
210
310
410
-- ,I, 510 6.0
kod
7.85
(d)
6.28.
4.71 N
3.14 .,=
1
.
5
7
1.0
2.0
'
~
0.~' - I .57 '
0.0
3.0
4.0
5.0
6.0
kod FIG. 10. (a) Magnitude, (b) phase of horizontal force, (c) magnitude and (d) phase of vertical force as a function of kod (for different values of W / d and h/d, 0 = 0).
700
A. G. ABuL-AzM 1.2
(a)
1.o 0.8-
;,
tt
0.60.4-
ii I
X
$ 4 *1
=_
0.20.0
--
_
i
•
_ l
•
20
lO
-
_.
|
_.
•
30
-
4'0
•
,'0
|
8'0
¢0
80
90
T H E T A (Deg.)
1.2" (b) ~"
1.0-
--
0.8"
-
=__
-
kod=0•5, W]d --0.5
0.6"
--÷--
k/l=0.5, W/d=l.O
0.4"
I ........
krd=l.5, W/d=l.O
0
10
20
30
40
50
/~- /
,
' " . - ~
0.0
,,( ./~/~,:~I. [/////\
"-'--4--~-.4."
0.2
"',\'~,)
k~=~.5, w/d=o•5
-
,
60
70
,
80
90
T H E T A (Deg.)
1.2-
(c) _
1 .o-
.117.-. . . . . . . . . . .~. . . . . ~
~
]
~
~
0.8- " - ' P " P - ' P -'P
0"6• ~e
+" -,o- 4.._+. ,4,. ~
,,,'/ /'~,
..."//A"
0.4.
-
0.2, ,.
0.0
.
.
.
.
-°
.'" 10
20
30
40
50
60
70
80
go
T H E T A (Deg.) FI~• 11. Magnitude of reflection and transmission coefficients (a) h/d = 0.25, (b) hid = 0.50 and (c) hid = 0.75, as a function of 0 (for different values of/cod and W/d).
Wide submerged breakwaters 0.8
(,.)
0.6
6 = 15 ° -----o--- 9 = 3 0 " .... t .... 8 = 45 °
¸
..O II"
"~0.
"lD%
a •" 0.4
701
~.,
!!
---z,--.
O = 60 °
.... Q " "
8=75"
.....
0 = O*
oo
•
.
0.2, .m !_..'-
• ,,~ 0.0,
~ ..':llm~
~ -~ =,'-~, %
"u.
12.-" ~ ~ - ~ : ~ " -
•
0.0
,
•
-
1.0
" •
=.=.
,
,
~
2.0
3.0
4.0
-
--
5.0
6.0
kod
785] (b)
!
-
I
tA !
4.71 "~'
i
! ~
,d~r'.,..O ' ~ r =_..ao.a°"~ '~,
]
,.i,~!l
s,, J 1.57 1 0.~!
0.0
!
110
210
:310
4.0!
5.0
6.0
koa P[o. 12. (a) Magnitude and (b) phase of the reflection coefficient as a function of/cod (for W / d = 1.0 and variable wave angles).
hid
= 0.5,
702
A . G . ABUL-AZM 0.8
(a) o,D
0.6 ¸
o e ' 4 " 41' ° 4 ° "° ' m " "° " '1' • "° ° ~
i,
;
"e..,e
!
m
"e.
0.4' ~-o-O-,I~I-o. a ;P 0.2"
!A
,
"..
I/,;.
~a~D~.ID
~
"u.
V~\
.-"""""~',~.
0.0 0,0
1.0
2.0
3.0
4.0
5.0
6.0
kod
I
j (b)
785 I
~I ~ =¢
'"t
eL
3.14
I
,11,,41
0.~0°A°
1.57
t"
0.00 / 0.0
i
!
i
!
!
1.0
2.0
3.0
4.0
5.0
6.0
kod FIG. 13. (a) Magnitude and (b) phase of the reflection coefficient as a function of kod (for h/d = 0.75, W/d = 0.5 and different wave angles); for legend, see Fig. 12.
Wide submerged breakwaters
703
1.o
(a) Full Solution Plane-Wave Approx. Long Wave Approx.
0.8 % % \
0.6" m
0.4"
% %% ,%
0.2'
0.0
,% %,
10
100
Lo/d
9.42 "t
(b)
7.85 "4
\
¢1
m
6.28
¢1
,,= 4.71
3
I
.
1
4
10
~
100
Lo/d FIG. 14. Comparison of (a) magnitude and (b) phase of the reflection coefficient as a function of Lo/d, using different theories (for h/d = 0.25, w/d = 0.5 and 0 = 0).
704
A.G.
AauL-AzM
1.0. •~
(a)
~,
" ... ~-
Full Solution - - -
0.8-
. ...........
.....
P l a n e - W a v e Approx. L o n g W a v e Approx.
\4
0.6
0.4
0.2
0.0 10
100
LJd
9,42 (b)
7.85 I
•=
i
6.28
1
¢I
4.71 1
"¢
3.14
1.57
0.~
. . . . . . . .
1'0
. . . . . . .
I00
Lo/d FIG. 15. Comparison o f (a) magnitude and (b) phase o f reflection coefficient as a function o f Lo/d, using different theories ( f o r hid = 0.75, w/d = 1.0 and 0 = 0).
Wide submerged breakwaters
705
e x p l a i n e d in t e r m s o f less i n t e r f e r e n c e with t h e free surface, as s h o w n in Fig. 14. A s t h e w a v e l e n g t h L o / d i n c r e a s e s , t h e l o n g - w a v e a p p r o x i m a t e results t e n d to h a v e a b e t t e r r e p r e s e n t a t i o n o f t h e full s o l u t i o n results, as s h o w n in Fig. 15. CONCLUSION A n e x a c t m e t h o d b a s e d o n e i g e n f u n c t i o n e x p a n s i o n t e c h n i q u e has b e e n utilized to d e t e r m i n e t h e e f f e c t i v e n e s s o f a w i d e , s u b m e r g e d a n d i m p e r m e a b l e b r e a k w a t e r in reflecting o b l i q u e waves. H y d r o d y n a m i c h o r i z o n t a l a n d v e r t i c a l forces h a v e b e e n c o m p u t e d o n t h e b r e a k w a t e r . U n d e r t h e a s s u m p t i o n o f l i n e a r w a v e s a n d p o t e n t i a l flow t h e o r i e s , t h e n u m e r i c a l results o b t a i n e d f r o m t h e p r e s e n t m o d e l h a v e s h o w n a g o o d a g r e e m e n t with t h e c o r r e s p o n d i n g e x p e r i m e n t a l results o b t a i n e d b y o t h e r i n v e s t i g a t o r s , p a r t i c u l a r l y for flatter waves. T h e t h e o r y has b e e n e x t e n d e d to o b t a i n t w o a p p r o x i m a t e solutions, n a m e l y t h e p l a n e - w a v e a p p r o x i m a t i o n a n d t h e l o n g - w a v e solution. T h e results o b t a i n e d f r o m t h e two a p p r o x i m a t i o n s h a v e b e e n s h o w n to give a g o o d e s t i m a t e for the full s o l u t i o n results in d e e p a n d shallow w a t e r , r e s p e c t i v e l y .
REFERENCES ABuL-AzM, A.G. 1993. Wave diffraction through submerged breakwaters. J. WatWay, Port, Coastal Ocean Engng, ASCE 119, 587-605. ABDUL KHADER,M.H. and RAL S.P. 1980. A study of submerged breakwaters. J. Hydr. Res. 18, 113-121. DALRYMPLE,R.A., LOSADA,M.A. and MASON,P.A. 1991. Reflection and transmission from porous structures under oblique wave attack. J. Fluid Mech. 224, 625-644. DALRYMPLE,R.A. and MAi~TIr~,P.A. 1990. Wave diffraction through offshore breakwaters. J. WatWay, Port, Coastal Ocean Engng, ASCE 116, 727-741. DAITATRI,J., RA~Ar~, H. and SnANKAR,J. 1978. Performance characteristics of submerged breakwaters. Proceedings of the 16th Coastal Engineering Conference, ASCE, Hamburg. DEAN, W.R. 1945. On the reflection of the surface waves by submerged plane barrier. Proc. Cambridge Phil. Soc. 4, 231-236. DICK, T.M. and BREBNER,A. 1968. Solid and permeable submerged breakwaters. In Proceedings of the llth Coastal Engineering Conference, ASCE, London, Vol. II, pp. 1141-1158. HALL,W.C. and HALL,J.V. 1940. A model study of the effect of submerged breakwaters on wave action. U.S. Army Corps of Engineers, Beach Erosion Board, Technical Memorandum, No. 1, May 1940. JEFFREYS,H. 1944. Note on the offshore bar problems and reflection from a bar. Great Britain Ministry of Supply, Wave Report No. 3, June 1944. JOHr~SON, J.W., Fucns, R.A. and Movasor~, J.R. 1951. The damping action of submerged breakwaters. Trans. Amer. Geophys. Union 32, 704-718. KOBAYASltl,N. and WORTANJO,A. 1989. Wave transmission over submerged breakwaters. J. WatWay, Port, Coastal Ocean Engng, ASCE 115, 662-680. MADSEN, P.A. 1983. Wave reflection from a vertical permeable wave absorber. Coastal Engng 7, 381-396. SABOr~C~,T. and GORES, O. 1992. Wave propagation over sloping bottom and submerged breakwaters. In Proceedings of the 2nd International Marina Technical Conference, Southampton, U.K., pp. 185-195. SAV.Pr,AVA, T. and ISAACSON,M. 1981. Mechanics of Wave Forces on Offshore Structures. Van NostrandReinhold, New York. SILVESrER,R. 1976. Headlands defence of coasts. In Proceedings of the 15th Coastal Engineering Conference, pp. 1394-1406. SOLLITF,C.K. and CROSS, R.H. 1972. Wave reflection and transmission at permeable breakwaters. Department of Civil Engineering, School of Engineering, MIT, Report No. 147, March 1972. SuLisz, W. 1985. Wave reflection and transmission at permeable breakwaters of arbitrary cross-section. Coastal Engng 9, 371-386. TWERSKV, V. 1962. On the scattering of waves by the infinite grating of circular cylinders. IRE Trans. Antennas Propag. 10, 737-765. WIEGEL,R.L. 1964. Oceanographical engineering. In Theoretical and Applied Fluid Mechanics. PrenticeHall, Englewood Cliffs.
706
A . O . ABut:Azr~
APPENDIX In this Appendix, explicit expressions are given for the z-integrals appearing in Equations (11), (13) and (16): sinh 2kod + 2kod
4ko sinh2kod q=0 Nq = sin 2kqd + 2kqd q >- 1 4kq sinh2kod [- sinh 26o(d-h ) + 2f3o(d-h ) Sp = [ sin 213p(d-h) + 2~e(d-h) t - - 4~3ps i n h 2 ~ ( d - h ) sinh
Too =
(A1)
(A2) p -- 1
(koz+fgo(z-h)) sinh (koz-~o(z-h)) i" O-1 2(ko+fSo) ~2(ko-~o)
Toq = {f3ocoskqzsinh~3o(z-h)(f32o + k~) + kqsinkqzcosh~3o(z-h)(d_h) --
(A3)
i • D -l ,
q -> 1
(A4)
Tpo = {f3ocoshkozsinf3p(z-h)+(f3zp + kz)k°sinhk°zc°sf3e(z-h)(d-h) i • D - ' ,
p -> 1
(A5)
Isinkqz+f3p(z+h)) sin(kqz-f3p(z-h)) i Tpq [ 2(kq-+~p) + 2(kq'~t,~ " D-I' =
and D = sinh
kod sinh fSo(d-h).
P'
q-> 1
(A6)