Transmission of random wind waves through perforated or porous breakwaters

Coastal Engineering, 1 (1977) 63--78 © Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands

TRANSMISSION OF RANDOM WIND WAVES THROUGH PERFORATED OR POROUS BREAKWATERS

S.R. MASSEL and C.C. MEI

Institute of Hydroengineering, Polish Academy of Sciences, Gdansk (Poland) Department of Civil Engineering, Massachusetts Institute of Technology, Cambridge, Mass. (U.S.A.) (Received October 2, 1976; accepted December 12, 1976)

ABSTRACT Massel, S.R. and Mei, C.C., 1977. Transmission of random wind waves through perforated or porous breakwaters. Coastal Eng., 1: 63--78. An analytical theory for random waves impinging on a dissipative breakwater is presented. From JONSWAP spectrum which was measured in depths including those of coastal engineer. ing interest, it is pointed out that there can be cases where most of the energy is contained in the band of wavelengths long compared to depth. A linearized long-wave approximation is then used. For a perforated breakwater or a porous breakwater of the crib type, a stochastic equivalent linearization technique is applied to treat the quadratic loss at the breakwater. The random-wave scattering problems are then reduced to simple harmonic ones whose solutions are known. Statistical reflection and transmission coefficients are defined and numerical examples are given.

INTRODUCTION

In the past decade, theoretical and experimental efforts have been made to estimate the properties of t w o types of dissipative breakwaters. Perforated and slotted breakwaters have been studied in Hayashi et al. (1966), Richey and Sollitt (1969), Mei et al. (1974), and rubble m o u n d breakwaters in Kondo and Toma (1972), Sollitt and Cross (1972), Madsen (1974), and also by C.C. Mei in an unpublished note. In all these works the incident waves are assumed to be sinusoidal and the effect of damping is represented empirically by a term quadratic in the local velocity. In the analysis it is customary to use an idea due to H.A. Lorentz and replace the nonlinear term by a linear one so as to give the same average rate of dissipation. With proper choice of the empirical constants, the theories seem to be reasonably supported b y laboratory experiments. In natural environment wind-generated waves are seldom sinusoidal, and a more realistic description must involve the use of statistics. In this paper we shall apply a spectral theory to wave scattering b y perforated, slotted or

64

porous breakwaters. Use will be made of the statistical version of the equivalent linearization technique which is well-known in control theory (Booton, 1953}, random vibrations (Caughey, 1963) and in ship motions (Kaplan, 1966). In particular we shall use the shallow-water approximation which is meaningful if the energy of the incident waves is largely contained in the long-wave range of the spectrum. To show that this approximation can be of practical interest we first c o m m e n t on the available spectral data. Spectral theories for wave scattering are then presented. Analogous to simpler linear systems, we find that the problem of random-wave scattering is reduced to that of the sinusoidal wave scattering. Statistical transmission and reflection coefficients are introduced. Some numerical results are given at the end. A BRIEF R E V I E W O F T H E K N O W N

WAVE

SPECTRA

The random incident wave propagating over constant depth is assumed to be Gaussian with zero mean and is represented by the following FourierStieltjes integral: oo

[i(x,t) =

f

dA(w)e i(t°t-kx)

(1)

where co: = g k tanh kh. For a stationary and homogeneous process, the amplitude spectrum dA satisfies: E[dA(co )dA* (co')] = Si (co)8(60 - co ')dtod¢o ' where E[ ] represents the statistical average, ;~i the power spectrum of the incident wave defined for all real co and 5 the Dirac delta function. It is recalled that, since [i is real, Si (co) = S ~ ( - w ). The auto-covariance p i ( r ) and Si (co) are the Fourier transforms of each other. Define Si (co) = 2 S i (co) for co > / 0 one has: E t ~ i ( t ) ~ i ( t + r)] =

pi('f) = 2 f Si o

Si(¢O) = - ~ i ( ¢ 0 )

c o s ¢oTd~ - ; o

pi (7")COS ¢ordr

=

S i cos

for co > 0

¢o~'d¢o

(2)

(3)

o

Furthermore the standard deviation o i2.is a i2 = pi(O). At present available data on the sea spectrum is mostly for deep water. A typical one is the well-known Pierson-Moskowitz (1964) (P-M) spectrum which is valid for a fully aroused sea. In shallow water, results of extensive records b y the U.S. Army Coastal Engineering Research Center (CERC) (Thompson, 1974) show that the measured wave spectra differ significantly from the P-M

65

spectrum; a b o u t half of the records give higher spectral peaks than the latter. Unfortunately, the accompanying meteorological data for these records, i.e., fetch, wind velocity and duration, are n o t adequately reported. Spectral analysis of the Norwegian coast ( H o u m b et al., 1974) also shows similar deviation. The most extensively measured and analyzed effort is the JONSWAP spectrum, which was recently taken in a depth range of 5--40 m (17--131 ft.). The JONSWAP spectrum can be approximated b y the following analytical expression (Hasselmann et al., 1973, eq. 2.4.1): Si

(~)

=

~g2(2~) -4

a = Oa

CO -s

for co/cop < 1,

exp

5 4

and

CO a = Ob

7 e x p [ ( w / w v - 1)5/2o 2]

(4)

for CO/COp> 1

which involves five empirical parameters COp, a, 7, aa and Oh, where a = Phillips constant depending on fetch; COp = frequency of spectral peak; T = peak enhancement factor; and a = spectral width parameter. The factor before 7 is the P-M spectrum. Define a dimensionless fetch parameter: = gZ/U~, o

Z =

fetch

(5)

where U~0 is the wind speed measured at 10 m above the mean sea level; the empirical relations according to JONSWAP measurements are: a = 0.076 ~:-o.22 COp _ 3.5 .~-0.33 2~ 7 -- 3.3,

Oa = 0.07,

(6) Ob = 0.09

It is the opinion of H o u m b et al. (1974) and Thompson (1974) that the mean JONSWAP spectrum is representative of moderately shallow sea. Strekalov et al. (1972) reported the spectrum measured in Barentz Sea, Black Sea and Kaspian Sea, the depth range being 20--500 m (65--1640 ft.). The wind speed is 1 0 - 2 0 m/s ( l g . 5 - - 3 9 knots). The fitting formula is:

S(co)~ / ~ - 0.38 exp[-35(~ - 0.8)] + 0.22(~o)-Sexp[-l.34 ~o-8]

(7)

where ~ = mean frequency, ~ = c o / ~ , / ) = mean wave height. In Fig. 1, spectra b y P-M, JONSWAP and Strekalov et al. are compared for the following normalized shape factor: S

= (2~)4~-' g -2

S

(8)

It can be seen that the spectra b y JONSWAP and Strekalov et al. are qualitatively similar b u t differ significantly from the P-M spectrum..Simple spectra

66

for coastal waters have n o t been very well established. Darbyshire (1961) gives some information on the shallow-sea spectrum b u t his formula is t o o complicated for practical purposes. Finally, Druet et al. (1972) reported that for a shallow water (h ~ 5 m) the spectrum can be described by eq. 7 with revised values of the empirical coefficients. It should be n o t e d that in all these spectra the dependence on water depth has n o t been given explicitly. In this paper, the JONSWAP data will be used for its extensiveness as the basis of discussion. THE LONG WAVE

APPROXIMATION

In principle the sea spectrum contains all frequencies and wavelengths. However, most of the energy resides in a short range where the wavelength is quite long compared to the depth of the site of many breakwaters (h < 20 m say). For JONSWAP spectrum, most of the energy lies in 0.6 < co/cop < 2. Now for k h < 0.5 where k = co/x/g~,, the long wave equations:

--+h--=0

a~

au

re+g--=0

au

a~-

at

ax

at

ax

(9)

are fairly good for sufficiently small amplitudes. Combining with co/cop < 2 one gets the following rough criterion on the depth for applying the long wave approximation: h < g/(4cop) 2

(10)

For instance, let X = 900 km and U10 = 20 m s -1, the peak frequency cop is

1.0' .8

.~

.7

I. 7 f "'~'.

•6

'

.5-

.I"

/

I/ / I/ i I ,'

l/ " //I

.3-

I

2_ .I 06

I 0.8

I I.O

I 12

~/O.,p

I 1.4

I 1.6

I 1.8

I~

2.0-

Fig. 1. T h e n o r m a l i z e d s h a p e f u n c t i o n o f t h r e e m e a s u r e d s p e c t r a ..... - P i e r s o n - M o s k o w i t z ; . . . . eq. 7.

-JONSWAP;

67 0.38 r a d s - ' . H e n c e t h e d e p t h s h o u l d be less t h a n 14 m (42 ft.). Since w i t h o u t t h e long wave a p p r o x i m a t i o n t h e t h e o r y is m a n y t i m e s m o r e c o m p l e x (Sollitt a n d Cross, 1 9 7 2 ) it m a y be a practical p o l i c y t o a d o p t a long wave a p p r o x i m a t i o n even f o r slightly greater d e p t h . L e t us n o w t r e a t specific p r o b l e m s . TRANSMISSION THROUGH A PERFORATED OR SLOTTED BREAKWATER

L e t it be a s s u m e d t h a t at x = 0 t h e r e is a p e r f o r a t e d o r s l o t t e d b r e a k w a t e r o r a b r e a k w a t e r m a d e u p o f closely-spaced piles. F o r illustration n o solid b a c k wall is a s s u m e d so t h a t t h e i n c i d e n t waves f r o m x ~ --= can be transm i t t e d t o x ~ +~. With a b a c k wall at x = L, t h e p r o b l e m can b e t r e a t e d similarly w i t h o u t d i f f i c u l t y . On t h e i n c i d e n c e side x < 0, t h e r e is an i n c i d e n t wave d e s c r i b e d b y eq. 1 w i t h the c o r r e s p o n d i n g v e l o c i t y ui = V ~ h [i. T h e t o t a l wave field is:

~i = ~i + i R(CO )ei(c°t +kX)dA(CO)

(11)

Ul = Ui --

(12)

R(co )ei(c°t+kX) dA(CO )

with:

CO/k =

(13)

On t h e transmission side x > 0 t h e wave field is r e p r e s e n t e d by: ~2 =

T(co)ei(°~t-kX)dA(co)

u2 =

~2

(14)

--oo

T h e q u a n t i t i e s R a n d T s h o u l d be f o u n d b y invoking t h e f o l l o w i n g b o u n d a r y conditions: ul(O,t) = u d O , t ) =- Uo(t)

(15)

_~'dO,t) - ~2(O,t) =

(16)

u0[u0l

T h e n o n l i n e a r b o u n d a r y c o n d i t i o n o f eq. 16 will n o w be r e p l a c e d b y t h e linear o n e :

fe

~l(0,t) - ~2(0,t) = Uo COp

(17)

in such a w a y as t o m i n i m i z e t h e e r r o r :

f 2g

e =--UolUol

--~

fe Uo COp

(18)

68 in the statistical mean-square sense, i.e.: a

afe

E[e 2] = 0

(19)

The result is:

fe = fOop E[u2olu01l 2g E[u~]

(20)

Compared to the deterministic case of sinusoidal waves it is recognized that the statistical average is used to replace the time average. By virtue of the assumed linearity of the equivalent problem, it is reasonable to assume that uo(t) is also Gaussian with the probability density function: 0 -o -1- exp

p.o(U0)

(21)

/

2 = E[u~] is the variance of u0. It is well known that: where Ouo 2~/-2 3

E[u~ luol ] =--~- OUo so that: r - -

fe

V 2 fCOp O'Uo

(22)

g

(see e.g., Kaplan, 1966). By writing:

Uo(t) = ; Uo(co)eic°tdA(co )

(23)

--oo

it is easy to show that R, T, U0 are just the response to the linear, sinusoidal and deterministic problem. The result is simply: u0=V-h

1

fe l+'-2kph

T=Vr-~Uo

R=I--T

(24)

which are independent of co. Also because of linearity one has: Si 2OJp/ Since the denominator above does not depend on co,

aUo is easily found to be:

69

y

OUo

1

0

2

+

f

f

+f___ 2

0

g

2COp

(26)

2COp

Combining eq. 26 with 22 one obtains: ] g ~ x / 1 + 2fl -- 1 oi

Ou o

(27)

v-vh

and:

y•2foi

fl = V-~ ~

(28)

The spectra of the transmitted and the reflected waves are: S T ( W ) = $3(co) = S i ITI 2 = Si

S R (C0) = S i IRI 2 = S i

(1 + fl

-

(V~ + 2fl - 1):

~2 I~/1-V~)2

~2

(29)

(30)

On the hypothesis that the incident wave is Gaussian with a narrow-banded spectrum, the m a x i m u m amplitude is Rayleigh-distributed and: E[AI

=

(31)

oi

N o w since o~. = a~ = ITI 2 and o k = IRb~, one may introduce the statistical transmission and the reflection coefficients as follows: oT

K T =--

= T =

+ 2fl

-

1

1 1 + / 3 - - ~¢/1- + 2ill

oR

KR =

~

=R =

ai

(32) (33)

/3

which are identical in form to those of the deterministic problem (Terrett et al., 1968) where (~ is defined to be 4 A f / 3 1 r h in contrast to eq. 28. Thus the t w o theories are the same if one defines an equivalent incident amplitude Ae: A e =4-3~ ~

ai = ~3E [ A ]

(34)

after using eq. 31. We remark that the numerical factor 3/2 above is the consequence of Gaussian narrow bandedness. If instead we assume the incident wave to be an ensemble of sine waves with constant amplitude A and frequency b u t with uniformly distributed phase, it may be shown that E [ A ] = A i = X/~-ai, which implies A e = A i b y the first equality of eq. 34.

70 Lastly, the spectrum and the standard deviation of the total wave ~, on the side x < 0 are: ~

1 + ~ + ~ cos

~-

01=0 i

-

(35)

Si(~)

I + ~ ) + S(w)cos

d

(36.)

0

These are the quantities which are more directly measurable on the incidence side than S R a n d OR. TRANSMISSION THROUGH A RECTANGULAR POROUS BREAKWATER The breakwater is assumed to occupy the space Ix l < I. Outside the breakwater, Ix l > l, eq. 9 holds. The porous breakwater is assumed to be macroscopically homogeneous so t h a t the following equations are valid: a~2 h 3~u2 = 0 ax

at

C au~_

1 ap2

-

n at

(37)

au 2 - bu21u~l

(38)

pax

where n = porosity and C the inertia coefficient. These equations have been used for regular waves transmitted through a porous breakwater. As is discussed throroughly by Madsen, the inertia coefficient may be approximated by unity for m a n y practical circumstances. We shall therefore take C = 1. To simplify the analysis the damping terms will be replaced by an equivalent linear term so that eq. 38 is replaced by: -

1 a u2 --=

1 ap~

fe

nat

p ax

n

¢OpU2

(39)

The details of the equivalent friction coefficient will be discussed later. Suffice it to mention that fe is a constant and ~0p is a reference frequency. Now the wave field to the left of the breakwater is described by: ~1 = } dA(co){exp i[¢~t - k ( x + / ) ] + R exp i[¢ot + k ( x + / ) ] }

(40)

--oo

and: u, =

dA(co){exp i[~ot - k ( x + l)] - R exp i[cot + k ( x + l)l }

(41)

71 which consists of the incident wave and the reflected wave. To the right the transmitted wave is described by: ~s =

J

TdA(co) exp i [ c o t -- k ( x - / ) 1

us =

~s

(42)

-oo

Inside the breakwater the solution is assumed to be: ~2 = f

dA(CO)Z(x,CO )e i°~t

u: = S dA(co)U:(co,x )e i°~t

--oo

(43)

--oo

It is easy to show that Z, U2, T and R, are exactly the simple harmonic responses with R and T being the reflection and transmission coefficients. In particular from eqs. 37 and 43 Z and U satisfy the following ordinary differential equations : dU2 icoZ + h "

-gn = O, a n d

dZ

(44)

[72 =

dx

(ico 1 - i f e ~

p ) dx

which can be combined to give: dx 2 + p2Z = 0

with:

#2 = k 2 I

(45)

ife

In the deterministic case of sinusoidal incident waves, since most breakwater thickness is comparable to the water depth it is possible to use a perturbation analysis for small k l so that errors can be easily ascertained. The same analysis can be applied here if most of the energy of the incident-wave spectrum is in the long-wave range. Thus to the order kp l the approximate solutions axe: Z

= C - i D p x + 0(kp l) 2

U2 = g_k_k l ( D - i C t z x ) + 0(kp l) 2

(46)

co~

where: a =

np

(47)

1 -- ire

Use has been made of the fact that/~l _-< 0(kp l) for such a spectrum. By matching ~ and u at the interfaces x = el the coefficients R, T, C and D can be determined with the following result:

R =

a + it~t(1

1 - ire co'--p- n 2 co

1)

ilzl(ct 2 -+

a 2)

+ O(kp 02 --

+ O(kp 02 1 - - ire cop + n co

--i-- n kl

(48)

72

cy

T=

(II+ i/.ll(l + (Y2)

a+

c _

1

+ O&)1)2 =

+ O&1)2

da)

(59)

a + ipl(l + a2)

-

a2

D=

(Y +

(49)

(51)

ipZ(l + ~2’)

These results are consistent with those of Sollitt and Cross (1972). Note that the term D is associated with the spatial average of U,:

0,

=3

U,(x)dx

=

Ef +O(k,1)2 E

Ii

z

-1

1

+

+O(kp1)2

(52)

"Plfe

n Let us now turn to the equivalent friction coefficient fe. In the deterministic case, one may choose fe such that the total rate of energy dissipation, integrated ouer the breakwater width and averaged over a period is unchanged. Alternatively one may minimize the mean square error when the mean is taken over time as well as space. For random waves one simply replaces the time average by the stochastic average. Thus let the error be: f(x,t)=au2+bu21u21

feap

--u

n

2

Upon minimizing the mean square error: 1

l

s

2E-I

E[e’]dx

(54)

one gets: 1

b s E[u;lu21]& fewpza+ n

-I 1

(55)

.‘E[u;]dx _I-I ’ Assuming again that u2 is Gaussian one must have:

(56)

73 %2 is so far a function of x. By definition a 2U2 is:

o 2U2 = E[u~] =

ff

E[dA(co)d*A(co')]e i(w-¢°')tU2(x,~o ) U~*(x , ~ ')

--oo

f

=

IU2(x'c°)12'Si(c°) de° =

--ao

Ig2(x,w)12Si(co)dco

(57)

0

F r o m eqs. 46 and 51 it is straightforward to prove that: 1 I , 2-T.( IU212dx = 10212 + O(kpl)2 1

~-~

l

(58)

tu213dx = 102t 3 + 0 ( % 0 2

(59)

It follows that:

(60)

Ou~ = o5~ + O(kpO 2 and that: feCOpn -

a+

~boa2

(61)

'[l+O(kp/)21

By virtue of lineariW of the equivalent problem:

~Si(¢o ) Sac(c°)=

(62)

(l + kplfe )2 n

.

and:

Oi Ou2 = V~ 1 4 kplfe

(63)

n Substituting eq. 63 into 61 one gets: n fe - 2kpl I--(1--%~) + V(l+/a~]12+8

2 blaih

]

(64)

The deterministic version of these formulas is formally the same if one replaces

i by .-~ A, and was obtained by Madsen using a more intuitive

argument than that used here, and by Mei.

74 It does n o t seem to have been pointed o u t that the case in the previous section is a limit of the porous breakwater for l -* 0 and 2bl ~ fo. It is easy to see that in this limit eq. 64 reduces to: kplfe _ 1 n

2

(--I + V~ + 2~)

(65)

where/3 is now defined in eq. 28. Substituting eq. 65 into 63, one obtains eq. 27. The spectra of the transmitted and the reflected waves are:

S T =$3 = ITI2Si

and

SR = IRI2Si

(66)

The statistical transmission and reflection coefficients are:

KT =

oT

; o

=

IR(w)12Zi(w)d6o

Oi

f

Si(W )d~

0 o

( uR KR=--=

lh

(67)

IR(co )12 S i ( ~ )d6o

o

°i

;

Si(co)dco

0

In addition the spectrum for the total wave on the incidence side x < - l is: S1(6o~) = S i

l

F2co(x + l) [ l + IR(6o)I 2] + 21R(co)lcos l-

8r(¢O)11 (68a)

where 8 r is the phase angle of the complex R(¢o): R(¢o) = IRle -iar

(68b)

NUMERICAL RESULTS FOR A RUBBLE MOUND BREAKWATER The semi-empirical theory in the previous section relies on the knowledge of the hydraulic properties of the breakwater material a b o u t which a good discussion is found in Madsen (1974). For homogeneous porous media, Ward (1964) proposes the following formulas for laboratory materials: a = v/K

b = Cf/N/r-ff

(69, 70)

where ~ = laminar kinematic viscosity, K = intrinsic permeability and Cf is a dimensionless coefficient. According to Arbhabhirama and Dinoy (1973) Cf may be expressed as:

75

Cf = 100

dm

(71)

where dm = model rubble diameter. To extrapolate for p r o t o t y p e rubbles, these authors suggested that the permeability should be magnified by the ratio: K d2 gm= 42

(72)

where K and d refer to the prototype. As a sample calculation we take from the experiments of Sollitt and Cross (1972), dra = 3.4 cm (1.37 in.), Kra = 2.973 X 10 -3 cm 2 (3.2 X 10 -6 ft.2). For a p r o t o t y p e rubble diameter d = 1 m (3.05 ft.} the permeability is estimated to be: K = Km(d/dm)

2 = 2.45 cm 2 (2.64 X 10 -3 ft. 2)

(73}

Taking the experimental value of n = 0.437 as in Sollitt and Cross (1972} for the prototype, we obtain Cf = 0.364 from eq. 71. Finally from eqs. 69 and 70 one gets: a = 5.31 X 10 -3 s -l,

(74a)

b = 0.232 cm -1

(74b)

Let the water depth be h = 15 m (49 ft.} and the breakwater thickness 25 m (82 ft.). Assume t h a t the wind velocity is U10 = 20 m/sec (39 knots} and the fetch X = 900 km (480 miles). The non-dimensional fetch is ~ = 2.28 × 104 from eq. 73. The frequency at the spectral peak is COp = 0.385 tad/ sec (from eq. 74b). The standard deviation o.:z may be f o u n d by integrating the area under the curve S i . But it can be more simply obtained from Hasselmann et al. (1973) who gave: ~ = ( 1 . 6 ) 1 0 -7 ~7

(75)

with: (76a)

.~2 = a2 g2 U~I~

o2= f s 0

d CO-2n

(76b)

From this and the assumed values of .~ and Ul0 we obtain 02 = (2.4 m) 2. As with regular waves the energy spectrum of the reflected (transmitted) waves increases (decreases) with frequency. For low frequency, calculations have n o t been made for CO/COp< 0.6 because the incident energy in the JONSWAP spectrum is negligibly small. The equivalent linear friction coefficient is fe = 0.494. Finally by performing the numerical integration in eq. 67 the statistical transmission and reflection coefficients are found, K T = 0.412 and K R = 0.563. The spectra of the transmitted wave and of the total wave at ,the

76

front wall x = - l o f the breakwater are s h o w n in Fig. 2. Fig. 3 s h o w s the freq u e n c y d e p e n d e n c e o n the quantities:

IR(¢o)l 2 = S R / S i IT(~)I 2

(77a) (77b)

ST/S i

=

The energy spectrum o f the reflected (transmitted) waves increases (decreases) with frequency. This behavior is c o n s i s t e n t w i t h the reflection and transmission coefficients in regular waves (Sollitt and Cross, 1 9 7 2 ; Madsen, 1 9 7 4 ) . We have calculated K R and K T for the f o l l o w i n g range o f parameters: d = 0.5 ~ 3 m; half thickness l = 1 5 - - 4 5 m; sea d e p t h h = 1 0 - - 2 0 m; U10 = 1 0 - 3 0 m/sec; X = 2 5 0 - 1 , 0 0 0 km; n = 0.4. D u e t o the small e x p o n e n t s o f .~ in eq. 6 the d e p e n d e n c e o f KR and KT o n X is weak. F r o m eqs. 69 and 70 a is very small (laminar viscosity) while: b = Cflx/-ffo:d 114 so that KR and KT do n o t change significantly with the rubble diameters d. We o n l y e x h i b i t a typical d e p e n d e n c e o n the breakwater width in Fig. 4; values for other U and h are quite similar. , 1.0

S i (W/Wp)

,R-

S~ (~/wp)

S i ( I.~/I,~p}

a?:

3

0.75

,}. 7 i!

0.5

2

t

0.2~

. St

/

#:-~,"-0 8

0.5 0.6

I

/~/P\ L

L

\.\

I

1.0

1.2

1.4

1.6

1.8

2.0

~ l wp

Fig. 2. S p e c t r a o f t h e i n c i d e n t (S i* ) a n d t r a n s m i t t e d (S~) w a v e a n d o f t h e t o t a l w a v e s p e c t r u m o n t h e r e f l e c t i o n side (S*) at x = --l.

77

J 0.5 Sr(w) 0.4 Q3 02 Si (~) 0.1 ,

02

I

0.4

~

I

,

I

0.6

,

I

0.8

,

1,0

I

1.2

,

I

1.4

,

] 1.6

,

I

1.8

,

I

,

, =

2.0

w loJ p

Fig. 3. Typical f r e q u e n c y

dependence

of the reflection and the transmitted

spectra.

1.0 0.8

KR

0.6 0.4

~'X~x

"~"~

X~x.,,.x

KT x ~ x-,--..X~x~x ~

~.~

0.2 0

I I

I 2

I 5

-

-g Fig. 4. S t a t i s t i c a l t r a n s m i s s i o n a n d r e f l e c t i o n c o e f f i c i e n t s ( K R a n d KT) as i n f l u e n c e d b y b r e a k w a t e r t h i c k n e s s a n d f e t c h : U~o = 2 0 m / s ; h = 1 5 m ; d = 1 . 0 0 m ; n = 0 . 4 . --.~ = 6.13 × 103 (X = 250 kin) - x - x - :~ = 2 . 4 5 × 104 ( X = 103 k i n )

CONCLUSION

In this paper, semi-empirical theories are extended for r a n d o m waves passing: (1) a perforated, and (2) a porous breakwater. These theories are approximate for the shallow coastal zone where the energy of the incident wave is concentrated in the long-wave part of the spectrum. The quadratic damping term is treated by the stochastic equivalent linearization technique. The statistical transmission and reflection coefficients are introduced in terms of the standard deviations and the wave spectra. Although a n u m b e r of assumptions has been incorporated in this t h e o r y the simplicity of the results may be valuable in practical applications.

78

ACKNOWLEDGEMENTS

S.R. Massel wishes to thank the Polish Academy of Sciences and the U.S. National Academy of Sciences for making possible his visit in the United States. During the period of this study C.C. Mei was supported b y a contract with the Office of Naval Research and a grant from the National Science Foundation.

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