Wave-induced mean flows in vertical rubble mound structures

Coastal Engineering 35 Ž1998. 251–281

Wave-induced mean flows in vertical rubble mound structures Inigo ˜ J. Losada

a,)

, Robert A. Dalrymple b, Miguel A. Losada

c

a

b

Ocean and Coastal Research Group, UniÕersidad de Cantabria, Departamento de Ciencias y Tecnicas del ´ Agua y del Medio Ambiente, AÕda. de los Castros sr n. 39005 Santander, Spain Center for Applied Coastal Research, Department of CiÕil Engineering, UniÕersity of Delaware, Newark, DE 19716, USA c UniÕersidad de Granada, ETSI de Caminos, C. y P., Campus de la Cartuja sr n. 18071 Granada, Spain Received 10 June 1997; accepted 31 August 1998

Abstract Waves impinging on rubble mound breakwaters and seawalls induce a mean flow within the breakwater, analogous to the so-called undertow within the surf zone. Here, using a plane wave approximation Ž kh - 1.5., a second-order problem is solved for an idealized breakwater with a rectangular cross-section to show the origin and the nature of the mean flow within the porous structure. The mean flow is expressed in terms of a mean stream function analytically derived, obtained based on the mass flux balance between the incident, reflected and transmitted waves. Furthermore, the evolution of other second-order magnitudes such as mean water level and mass flux is analyzed under different incident wave conditions, structure geometry and porous material characteristics. Results show that the evolution of the different mean quantities is controlled mainly by reflection and consequently depends highly on structure geometry and porous material characteristics. Furthermore, it is shown that the return flow is stronger with increasing mass flux decay. Some qualitative experiments to show the described mechanism are also presented. q 1998 Elsevier Science B.V. All rights reserved. Keywords: Wave in porous media; Porous structure; Mean flows; Return flow; Mean wave quantities

1. Introduction Flow within porous breakwaters is responsible for the transmission of wave energy through the structure. In addition, it may affect the stability of the structure, although )

Corresponding author. Tel.: q34-942-201810; Fax: q34-942-201860; E-mail: [email protected]

0378-3839r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 3 9 Ž 9 8 . 0 0 0 4 1 - 6

252

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this latter effect is not well-known. Models have been proposed for wave-induced flows within permeable breakwaters by a number of authors Že.g., Madsen, 1974; Sollitt and Cross, 1976; Sulisz, 1994; Yu and Chwang, 1994.. These models have been developed on the basis of linear theory for the wave motion outside and within the breakwater. Matching conditions between the fluid region and the breakwater permit, for instance, the prediction of the damping within the structure. Dalrymple et al. Ž1991. and Losada et al. Ž1993. have shown that the plane waÕe approximation Ži.e., considering the most progressive mode only. provides a good representation of the flow fields for relative water depths, kh - 1.5, where k is the wave number, defined as k s 2prL Ž L is the wave length. and h is the water depth. Recently, a number of numerical models for the analysis of wave and structure interaction have provided flow fields for realistically shaped structures ŽWibbeler and Oumeraci, 1992; van Gent, 1994; Losada, 1996; Losada et al., 1996.. In general, three lines of development can be clearly identified: potential theory models, one-dimensional models based on long-wave equations, and Navier–Stokes models. Each of them is based on certain assumptions and has its own limitations. A complete description of the theoretical background of the different models as well as main advantages and disadvantages and range of application can be found in van Meer Ž1994., van Gent Ž1994. and van Gent Ž1995.. Here, we examine the second-order mean wave-induced water levels and flows within a vertical porous breakwater. It will be shown that this is due to the rapid decrease in the wave Eulerian mass transport induced by the damping in the porous breakwater. The mean flow that takes place below the wave troughs is directed offshore, exiting the structure near its toe. In this paper, it will be shown that a porous breakwater, as well as any other dissipative medium, is mechanically equivalent to the surf zone and, therefore, is able to induce similar phenomena such as mean water level variations Žset-up and set-down. and a mean flow analogous to undertow, ŽSvendsen and Putrevu, 1996.. The final goal of the paper is to develop a simple theoretical model to explain the mechanisms described. The complete derivation of the depth- and time-averaged equations for the mean motion in porous media can be found in Losada Ž1996.. The paper is organized as follows. The basic equations for wave propagating in a porous medium ŽSollitt and Cross, 1976. are presented as a basis for the solution of the first-order problem and of the further extension to second-order. In Section 2, the equations are extended to obtain the mean water level, mean water velocity at the still water level and the Eulerian mass transport at a vertical semi-infinite, finite and backed by a wall vertical permeable breakwater. A special section is devoted to the mean stream function which is presented for the geometries previously described. In Section 2, some theoretical results are included in order to analyze the influence of incident wave conditions, structure geometry and permeable material characteristics on the mean water level and mean flows. Some qualitative experiments to better described the return flow follow, as well as conclusions. In order to better point out the mechanisms presented most of the mathematical details have been included in two appendices.

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2. Theoretical analysis The analysis for the first-order solution will follow the plane wave approximation of Dalrymple et al. Ž1991., hereafter referred to as DLM. This approximation represents the wave motion as a progressive wave train, and neglects the evanescent modes that exist near the interface between the breakwater and the water region. This assumption reduces the complexity of the problem considerably allowing us to focus directly on the physics of the problem. Furthermore, it has to be considered that most of our coastal structures are located in relative water depths, kh - 1.5, and, therefore, in the solution range. 2.1. Water region The water region occurs for x - 0. The waves are assumed to be incident from x ™ y` and the breakwater will start at x s 0 ŽFig. 1.. The water region will be characterized by the usual linear wave theory Že.g., Dean and Dalrymple, 1984.. The Laplace equation, obtained as a result of assuming irrota-

Fig. 1. Definition of the problem for the finite breakwater showing cross-section and plane wave.

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tional motion and an incompressible fluid, must be satisfied by a velocity potential f , or a stream function c , which describes the fluid motion in the water region.

= 2f Ž x , z . s 0 or = 2c Ž x , z . s 0.

Ž 1.

The bottom boundary condition requires no flow into the bottom at z s yh. Also, two free surface boundary conditions must be met:

hs

1 Ef g Et

Ž 2.

v2

Ef Ez

on z s 0,

y g

f s 0,

Ž 3.

where g is the acceleration of gravity and v Žs 2prT . is the angular frequency of the periodic wave motion Žwith wave period T .. 2.2. Breakwater region The motion of an incompressible fluid in the pores of a porous structure is described in terms of the seepage velocity vector, q s Ž u,w ., where u and w are the components in the x and z coordinate directions, and the pore pressure is p. These quantities are obtained by averaging over a finite volume, containing both the solid phase of the porous medium and the pores. The continuity equation is given as:

= P q s 0, where = s ErE x Ž. i q ErE y Ž. j q ErE z Ž. k. The equation of motion includes resistance forces described by a modified Forchheimer’s model and an additional term which evaluates the additional resistance caused by the added mass of discrete grains within the porous medium ŽSollitt and Cross, 1976; Hannoura and McCorquodale, 1978.: s

Eq Et

s y=

ž

p

r

/

q gz y f v q,

Ž 4.

where the fluid has density r . The coefficient f is a linearized friction coefficient that can be obtained using the Lorenz equivalent work criterion ŽSollitt and Cross, 1976. given as: T

fs

1

v

H0 HV

ž

´ 2n Kp

2

q q T

H0 HV

´ 3Cf

(

Kp

2

q

3

/

dtdV

Ž 5.

´ q dtdV

where n is the kinematic fluid viscosity, ´ is porosity of the structure’s material, K p is 2 the intrinsic permeability, C f is the turbulent friction coefficient and q s q P q. K p

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and C f are related to the type of porous structure considered and are taken as given. s is an inertial coefficient, defined as: ss1q

1y´

´

CM ,

Ž 6.

where CM is the added-mass coefficient of the grains. The parameters Ž s, ´ ,CM , K p ,C f ., characterizing the porous medium can be evaluated a priori experimentally or using analytical expressions ŽSmith, 1991; van Gent, 1995.. V in the integral stands for a finite volume of pores. Irrotational motion satisfies Eq. Ž4., as can be verified by taking the curl of it. Coupled with incompressibility of the pore fluid, the irrotationality condition permits the existence of a stream function and a velocity potential. The flow in the porous medium can be described by a potential, q s =f . Substituting the potential into Eq. Ž4. and integrating results in a Bernoulli equation within the porous medium. Finally, substituting the potential into the conservation of mass equation yields Laplace’s equation for f which must hold everywhere within the medium. The validity of this theory has been shown by several authors Že.g., Sollitt and Cross, 1976; Sulisz, 1994; Chwang and Chan, 1998., and the hypothesis and range of application has been experimentally analyzed in detail by Losada et al. Ž1995.. Therefore, it will not be discussed in this paper. Furthermore, it has been applied with success to the resolution of different engineering problems as it is pointed out in the introduction of this paper. 2.3. Mean quantities To the moment, the theory for wave interaction with permeable structures has been used to analyze several engineering problems with success. In this paper, the first-order solution is extended to consider second-order effects, especially mean water level variations, mass transport and mean flows. 2.3.1. The instantaneous and mean Bernoulli equations The phreatic surface within the breakwater is given by the Bernoulli equation evaluated at z s h , with pŽ x,h . s 0 and a Bernoulli constants 0 Žwithout loss of generality. ŽSollitt and Cross, 1976.. s

ž

Ef Et

q

Ž u2 q w 2 . 2

/

q gz q f vf s 0 on z s h

Ž 7.

A Taylor series expansion permits evaluating this expression at z s 0 Žretaining only terms up to second-order in wave steepness.. s

ž

E 2f

Ef Et

q

E zE t

hq

Ž u2 q w 2 . 2

/

q gh q f vf q fwh q . . . s 0 on z s 0

Ž 8.

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Using only the linear terms Žthe first, fourth and fifth. and substituting for f yields the linear displacement.

hsy

1 g

Ef s

Et

q f vf on z s 0

Ž 9.

To determine the second-order mean water surface displacement, the Bernoulli equation is averaged over a wave period, neglecting third order terms and higher. This yields, ŽBattjes, 1974.: s hsy Ž 10 . Ž u 2 y w 2 . on z s 0. 2g 2.3.2. The mean kinematic free surface boundary condition At the phreatic surface, the fluid must satisfy the kinematic condition,

Eh Et

Eh qu

Ex

s w on z s h

Ž 11 .

which guarantees that the fluid moves with the surface. Expanding to still water level, z s 0, gives:

Eh Et

E 2f

Eh qu

Ex

swq

E z2

h on z s 0.

Ž 12 .

Taking the average over a wave period results in the mean kinematic free surface condition,

E 2f

Eh u

Ex

swq

E z2

h on z s 0.

Ž 13 .

Using the continuity equation, this can be rewritten as: ws

Euh Ex

on z s 0,

Ž 14 .

which indicates that w is a mean vertical velocity at z s 0, the still water level, which is due to changes Žin the x direction. in the Eulerian mass transport, ruh, occurring between trough and crest levels of the waves. The decrease in transport is due to the damping of the waves in the breakwater and provides the source for the mean flow within the structure. 2.4. Applications As a first and simple example, application to three different geometries of vertical permeable structures will be presented. These examples will help to better understand the influence of incident wave conditions, structure geometry and permeable material characteristics on the variations of the mean water level and mass transport induced by permeable structures.

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The solution of the second-order magnitudes requires the complete solution of the first-order problems considered. The latter can be found in Appendix A, summary of the eigenfunction expansion solution presented in DLM. 2.4.1. Semi-infinite breakwater To determine the second-order mean water surface displacement, Eq. Ž10. is used. Substituting for the velocities, using the potential Eq. ŽA.2. in Appendix A, and time-averaging Žremembering the complex nature of the functions. yields:

hsy

sa2 < A < 2

g
2

½ž /

5

y v 2 < s y if < 2 e 2 Q I x ,

v

4g

Ž 15 .

where Q s 'K 2 y l2 s Q R q iQ I , l s ksin u and u , the wave angle of incidence. The constants K s K r q iK i and k are the complex and real wave numbers, respectively, solution of the dispersion equations in a porous medium and in the water Eq. ŽA.3.. The imaginary part Q I is taken to be negative in order to account for dissipation, a is the amplitude of the incident wave and A stands for the complex amplitude of the wave propagating inside the semi-infinite breakwater. For normal incidence, u s 08, l s 0 and Q I s K I , and therefore,

hsy

sa2 < A < 2 4g

g
½ž / v

2

5

y v 2 < s y if < 2 e 2 K I x .

Ž 16 .

The set-down in Eq. Ž15. is a minimum at x s 0 and decreases exponentially to zero for large x Žactually only several multiples of the skin depth.. The mean vertical velocity for a semi-infinite porous breakwater can be evaluated using Eq. Ž14. and substituting expressions ŽA.2. and ŽA.10. in Appendix A to obtain: ga2 < A < 2 Q I Ž sQ R y fQ I . q sQ I l

e 2 Q I x on z s 0. Ž 17 . v For normal incidence, Ž l s 0 and Q I s K I , Q R s K R ., this expression can be simplified to: ws

ws

ga2 < A < 2 K I Ž sK R y fK I .

v

e 2 K I x on z s 0.

Ž 18 .

2.4.2. Finite breakwater Using the potential f 2 in Eq. ŽA.11. and substituting for the corresponding velocities in Eq. Ž10., we obtained the following expression for the second-order mean water surface displacement:

hsy

a2 s 4g a2 s

q 2g

g

2

½ž / Ž ½ž / Ž v

g

v

< Q < 2 q l2 . y v 2 < s y if < 2

2

< Q < 2 y l2 . q v 2 < s y if < 2

5 5 Ž

q Ž A R BR q A I B I . cosQ R Ž 2 x y b . 4 e Q I b ,

< A < 2 e 2 Q I x q < B < 2 ey2 Q I Ž xyb . 4 A R BI y A I BR . sinQ R Ž 2 x y b .

Ž 19 .

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where A s A r q iA i and B s Br q iBi are the complex amplitudes of the waves incoming from left and right, respectively, inside the finite permeable structure. The mean water level expression for normal incidence waves can be obtained simply by assuming Ž l s 0 and Q I s K I , Q R s K R ..It is straightforward to show that Eq. Ž19. with B s 0, reduces to Eq. Ž15., as was expected. The corresponding expression for the mean vertical velocity is: ga2

Ž sQ R y fQ I . Q I  AA) e 2 Q I x q BB ) ey2 Q I Ž xyb . 4 q

Ž 20 .

Ž sQ I q fQ R . Q R  AB ) eyi Q R Ž 2 xyb . q BA) e i Q R Ž 2 xyb . 4 e Q I b q

Ž 21 .

sQ I l  AA) e 2 Q I x y BB ) ey2 Q I Ž xyb . 4 y

Ž 22 .

isQ R l  AB ) eyi Q R Ž 2 xyb . y BA) e i Q R Ž 2 xyb . 4 e Q I b

Ž 23 .

ws

v

which for normal incidence is reduced to ws

ga2

v

Ž sK R y fK I . K I  AA) e 2 K I x q BB ) ey2 K I Ž xyb . 4

q Ž sK I q fK R . K R  AB ) eyi K R Ž 2 xyb . q BA) e i K R Ž 2 xyb . 4 e K I b

Ž 24 .

where K R and K I are the real and imaginary parts of the complex wavenumber in the porous medium, K, and ) stands for complex conjugate. 2.4.3. Finite breakwater backed by an impermeable wall The expressions for set-down and mean vertical velocity are the same given for the previous case, but using the R, A, B given by Eq. ŽA.16..

3. The mean wave-induced flow outside and within the structure 3.1. Introduction The mean wave-induced flow can be conveniently defined by the mean stream function, C . Based on the irrotationality condition and assuming harmonic motion, the boundary value problem for the stream function in the fluid and the breakwater regions requires the Laplace equation to be satisfied as well as the kinematic condition ŽEq. Ž14.. that can be found using the linear velocity potentials on the fluid and breakwater regions. At the bottom, the no-flow condition has to be satisfied. Mathematically, the problem for constant depth can be written as:

= 2C Ž x , z ,t . s 0

½

jFxFg yh F z F 0

Ž 25 .

This equation is valid, for both the fluid and the porous medium. The constants Ž j ,g . represent the problem domain limits in the x direction.

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The free surface boundary condition is given by:

Cx s V j z s h where

½

V 1 s 0 in a fluid domain V 2 s w in a porous domain,

Ž 26 .

where w is the mean vertical velocity that can be calculated using Eq. Ž14.. The bottom boundary conditions can be expressed as:

Cx s 0 z s yh Ž 27 . Furthermore, a condition of finiteness in x has to be imposed to close the problem. In order to find the global solution, a mass flux balance between the different regions has to be established. In front of the structure, mass flux results from the interaction between incident and reflected waves and can be expressed in terms of the reflection coefficient, R, whereas leewards of the structures the mass flux is a function of the transmitted wave. Due to the dissipation induced by the permeable structure, a reduction of the horizontal mass flux is induced that can be transformed into a vertical mass flux, finally exiting through the front face of the structure. The complete solution to each of the boundary value problems, which correspond to the geometries of the structures considered in the previous section, may be found in Appendix B including the mass balance as well as the matching conditions at the interfaces which guarantee the continuity of the solution. It will be shown that for the different cases studied, the stream function describes a return flow towards the sea of the mass transport carried into the structure by the wave motion. The flow develops due to the diminishing mass transport with distance into the structure. In this section, the expressions of the stream function for each of the cases considered are summarized. 3.1.1. Finite breakwater case Seaward water region: `

C 1 Ž x , z . s Ž U1 y U3 . Ž z q h . q

Ý Cn sin ns1

np h

Ž h q z . e np x r h .

Ž 28 .

Porous structure region:

C2 Ž x , z . s

1 a2 g 2 v a2 g q

v

Ž sK R y fK I .

sin2 K I Ž h q z . sin2 K I h

Ž sK I q fK R . e K I b

 < A< 2 e2 K

sinh2 K R Ž h q z . sinh2 K R h

= w A R BI y A I BR x q sin K R Ž 2 x y b . yU3 = w A R BR q A I B I x 4 q Ž z q h. . ´ Leeward water region: ` np C 3 Ž x , z . s Ý Dn sin Ž h q z . eyn p Ž xyb.r h . h ns1

I

x

y < B < 2 ey2 K I Ž xyb . 4

 cos K R Ž 2 x y b .

Ž 29 .

Ž 30 .

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260

The coefficients Cn and Dn , are given by: Cn s

´ a2 g Ž sK R y fK I . Ž AA) y BB ) e 2 K I b . 4 K I2 cos np v np q

4 K I2 y Ž nprh .

2

8 e K I b Ž sK I q fK R . K R2 cos np 4 K R2 q Ž nprh .

2

=  cos K R b w A R BI y A I BR x y sin K R b w A R BR q A I B I x 4 ,

Dn s

Ž 31 .

´ a 2 g Ž sK R y fK I . Ž AA) e 2 K I b y BB ) . 4 K I2 cos np v np q

4 K I2 y Ž nprh .

2

8 e K I b Ž sK I q fK R . K R2 cos np 4 K R2 q Ž nprh .

2

=  cos K R b w A R BI y A I BR x q sin K R b w A R BR q A I BI x 4 .

Ž 32 .

As it can be seen, the stream function expressions are a function of the incident wave amplitude, a, the complex amplitudes inside the structure, A and B, water and porous region wavenumbers, k and K, and the characteristics of the permeable medium given by ´ , s and f. Therefore, the first-order solution is required to be known. U1 and U3 are defined as M1rr h and M3rr h, respectively, where M1 s r 12 g Ž a2 krv .Ž1 y RR ) . and M3 s r 12 g Ž ka2rv .TT ) represent the mass flux in front and leewards the structure and are expressed in terms of the reflection, R and transmission, T, coefficients. 3.1.2. Semi-infinite breakwater The solution to the stream function can be found from Eq. Ž29., simply by assuming B s 0, T s 0 and U3 s 0. The reflection coefficient R and the complex amplitude A have to be determined using the first-order solution or a semi-infinite breakwater in Appendix A. Seaward water region: ` np C 1 Ž x , z . s U1 Ž z q h . q Ý Cn sin Ž h q z . e np x r h Ž 33 . h ns1 Breakwater region:

C2 Ž x , z . s Cn s

ga2 < A < 2 Ž sK R y fK I . sin2 K I Ž h q z . 2 v sin2 K I h

4´ a2 g < A < 2 Ž sK R y fK I . K I2 cos np

v np

4 K I2 y Ž nprh .

2

.

e2 K I x ,

Ž 34 . Ž 35 .

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3.1.3. Finite breakwater backed by an impermeable wall The expressions for the stream function are the same given for the finite breakwater case, but using the R, A and B given by Eq. ŽA.16.. Furthermore, U3 is taken to be zero since there is no mass flux through the wall.

4. Results In Fig. 2, the evolution of the non-dimensional mass flux in the direction of wave propagation is shown for a finite permeable breakwater. The results correspond to three different porous materials with characteristics given in Table 1. The three materials selected are used to show the sensitivity of the mass flux to the material hydraulic characteristics. Parameters are calculated using the empirical formulations of van Gent Ž1995.. The breakwater geometry is given by brh s 1, with h s 1 m. The wave conditions are given as the wave amplitude a s 0.05 m, a relative depth kh s 0.29 and the angle of incidence u s 308. For each of the given materials, a linearized friction coefficient, f, is evaluated using Eq. Ž5.. Results show that for all three materials, the mass flux decreases along the structure. It can be observed that the most important parameter controlling the mass flux is the intrinsic permeability, K p and indirectly the friction coefficient, f. Increasing, K p results in a larger transmission and lower reflection and therefore, material Ž3. is able to transmit a larger mass flux. Furthermore, increasing the friction coefficient results, in

Fig. 2. Mass flux evolution along a finite breakwater in the direction of the x axis for three different porous materials.

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Table 1 Permeable material characteristics Material

D50

´

Kp

Cf

1 2 3

5 cm 10 cm 15 cm

0.39 0.45 0.50

6.0)10y7 m2 3.0)10y6 m2 1.1)10y5 m2

0.15 0.11 0.10

this case, in higher reflection and, therefore, material Ž1. Ž f s 4.56. presents the lowest mass flux in front of the structure, since part of the mass flux is due to the reflected wave which travels in the negative x-direction. It can be observed that at the interfaces, the mass flux continuity is satisfied. It should be also pointed out that the mass flux decay inside the structure increases with f. In Fig. 3, the non-dimensional mass flux at a finite breakwater with a backwall with brh s 2.0 in a 1-m water depth is shown. Only one porous material is considered; however, several angles of incidence are used in order to analyze the influence of wave obliqueness. The no-mass flux condition at the backwall is satisfied for every angle of incidence. Furthermore, it can be seen that increasing obliqueness results in an increasing mass flux in the direction of the x-axis in front of the structure, since reflection is reduced with increasing obliqueness and, consequently, the part of mass flux associated with the reflected wave is small. This result is consistent with the result of the previous case analyzed. Notice that as the waves propagate inside the structure, the mass flux is reduced until it turns to zero at the backwall. It is also shown that increasing f results in a higher mass flux gradient inside the structure.

Fig. 3. Mass flux evolution along a finite breakwater with a backwall in the direction of the x axis for a given porous material and different angles of incidence.

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The variation of the mean water level at a finite porous breakwater is presented in Fig. 4. The breakwater and wave characteristics are given by the dimensionless width brh s 8, kh s 0.29, h s 1 m and a s 0.15 m. The water level is plotted for varying angles of incidence. The friction coefficient, f, is kept constant in order to make the comparison easier. In region Ž1., the Bernoulli constant C1 Ž t . s 0, therefore assuming that the reference level is located at the still water level. At the interfaces x s 0 and x s b, the mean water level is taken to be continuous, i.e., h1Ž0. s h 2Ž0. and h 2Ž b . s h 3Ž b .. These two conditions are used to evaluate the two additional constants, C2 Ž t . and C3 Ž t . , which correspond to regions Ž2. and Ž3., respectively. The figure shows that in front of the structure, the mean water level is modulated with a wave length, prQ R . This modulation results from the interaction between incident and reflected waves. For all the different angles considered, the mean water level is below the still water level Žset-down.. However, inside the structure, the mean water level presents a minimum at x s 0 and a relative maximum at x s b. This set-up is also slightly modulated, which can be clearly observed for the normal incidence case, where reflection from the lee face of the structure is the highest. The modulation inside the structure is controlled by prQ R and, therefore, affected by the porous material characteristics, breakwater geometry and incident wave conditions. The set-up is associated with the reduction in wave height due to dissipation inside the structure. Losada Ž1996. shows that for a permeable medium, the balance between mean water level variations and the gradients of the radiation stress, associated to the

Fig. 4. Mean water level variation in a finite breakwater for different angles of incidence and a single permeable material.

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seepage velocities, has to include several additional terms due to the spatial and temporal fluctuations inside the granular medium. Leewards of the structure, the main water level takes a constant value without any modulation. The value taken is imposed by the matching condition at the interface, h 2Ž b . s h 3Ž b .. Obviously, there is no modulation since region Ž3. is assumed to be semi-infinite and of constant depth, therefore, only transmitted waves exist in this region with no variation of the mean water level. Fig. 5 presents the same results for a finite breakwater with a backwall. The mean water level presents similar patterns as the previous case. The set-up inside the structure is even larger with a maximum clearly marked at the backwall, since the presence of the wall permits the piling of the water. In order to make an estimate of the order of magnitude of the mean water level variations in an actual structure, the following case is considered. Table 2 shows three different cases of actual wave conditions on the Northern coast of Spain. Wave height is considered to be H s 1 m. The actual structure could be located at a water depth h s 3 m, with a minimum width b s 12 m and built of mounds with W s 200 kg, D50 s 42 cm and ´ s 0.4. Using empirical expressions by other authors ŽSmith, 1991; van Gent, 1995. the intrinsic permeability, K p and turbulent friction coefficient, C f can be calculated resulting in K p s 3.26)10y5 m2 and C f s 0.226. Results are shown in Fig. 6, where the free surface envelope, hmax and the mean water level, h variation have been plotted. For the three angles of incidence considered,

Fig. 5. Mean water level variation in a finite breakwater with a backwall for different angles of incidence and a single permeable material.

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Table 2 Incident wave conditions

NE N NW

T Ž s.

u Ž8.

kh

10 12 16

20 5 0

0.35 0.29 0.22

a set-down at an approximate distance of twice the breakwater width in the offshore direction can be observed. The magnitude of the set-down is of 7 cm for kh s 0.29. The magnitude of set-down is decreased while approximating the front face of the structure and turns to a set-up with a maximum of 1 cm at x s b. Inside the structure, the mean water level increases with a maximum overall variation of 7 cm for kh s 0.29. As expected, the longer wave, kh s 0.22 is the least affected by the presence of the structure which results in small variations of the mean water level. The positive or negative variation of the mean water level inside the structure depends on the geometric characteristics of the structure, as well as, the material properties and incident wave conditions. In order to analyze the mean flows inside the structure, Fig. 7 shows the stream lines’ patterns for a finite porous breakwater with brh s 1, kh s 0.34 and ´ s 0.5. The stream function results always correspond to normal incidence. The linearized friction coeffi-

Fig. 6. Free surface and mean water level variation in a finite breakwater.

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Fig. 7. Stream lines of the return flow for a finite breakwater. br hs1, khs 0.34 and ´ s 0.5.

cient, f, takes the following values: 0.05, 1, 6 and 12. In the vertical axis, the non-dimensional vertical variable zrh is plotted vs. xrb in the horizontal axis. The porous breakwater is located between 0 - xrb - 1 and y1 - zrh - 0.

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The admittance m s ´ QrŽ s y if . q in Eq. ŽA.9. relates the vertical velocity with the pressure at the interface. This parameter takes into account the hydraulic characteristics of the structure. Increasing friction results in diminishing admittance and, therefore, the cases considered correspond to < m < s 0.5, 0.33, 0.20 and 0.14. In order to account for dissipation inside the structure, a different parameter, k d , is defined; where k is the wavenumber in region Ž1. and d is called skin depth. The relative skin depth is defined as the distance over which the motion has decayed to ey1 . Increasing values of k d require a longer distance for wave dissipation. If k d < 1, dissipation takes place in a very short distance. The linearized friction coefficient considered correspond to the following absolute values of k d : 0.99, 0.67, 0.4 and 0.3. For all the cases considered, it can be observed that a return flow in the offshore direction exists. The stream lines’ pattern as well as the velocities, are highly dependent on m and k d . For increasing f and, therefore, increasing k d , the negative wave height gradient increases. The larger the gradient is, the stronger velocities can be observed at the front face of the structure. For < m < s 0.14 and < k d < s 0.30, a descending current is clearly visible at the beginning of the structure with a narrowing of the stream lines at the toe of the structure indicating that the net current is strongest. In Fig. 8, a qualitative plot of the horizontal mean velocity profiles for the previous case is presented. The figure indicates that for < m < s 0.5, the smallest friction examined, the influence of the presence of the structure on the mean velocities is almost negligible. However, decreasing admittance turns out in a higher return flow due to increasing mass flux decay at z s 0. For < m < s 0.14, the velocity profile presents its maximum curvature, even showing a net flux in the direction of wave propagation at the top of the structure. In order to have an order of magnitude of the return flow, the mean horizontal velocity at the toe of the structure has been calculated using the mean stream function expression. The mean velocity at the front face of the structure is given as: u 2 Ž 0, z . s y

a2 g

v

´ Ž sK R y fK I .

2 a2 g y

v

cos2 K I Ž h q z . sin2 K I h

´ Ž sK I q fK R . e K I b

K I  < A< 2 y < B < 2 e2 K I b 4

cosh2 K R Ž h q z . sinh2 K R h

KR

P  cos K R Ž 2 x y b . w A R B I y A I BR x qsin K R Ž 2 x y b . w A R BR q A I B I x 4 q U3

Ž 36 .

The results show that, for normal incidence, H s 1 m and T s 10 s and considering the breakwater geometry and porous material previously described for Fig. 6, u s 4.1 cmrs. This velocity is able to move the sediment in suspension in front of the structure; therefore, contributing to toe erosion. Fig. 9 presents the same results for the finite breakwater considered in the previous figure, but with a backwall. The patterns are similar, but in this case stream lines are steeper, especially close to the wall, in order to satisfy the no flux condition at it.

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Fig. 8. Mean horizontal velocity profiles in a finite breakwater. br hs1, khs 0.34 and ´ s 0.5.

Finally, in Fig. 10, the stream lines for the semi-infinite breakwater are shown. For the small admittance case, < m < s 0.5, stream lines are almost parallel to the bottom, pointing out that dissipation is small. As the admittance decreases, dissipation increases and the curvature of the stream lines is more evident showing an important return flow. Especially interesting is the case for < m < s 0.14 where from zrh s y0.2 to z s 0, the stream lines show a slight flow in the direction of wave propagation. However, from zrh s y0.2 to zrh s y1, the flow is turned towards the offshore direction resulting in a return flow associated with dissipation that tends to discharge through the lower part of the water column all the Eulerian mass flow in the direction of wave propagation at the upper part of the water column. This vertical structure is similar to the bore and undertow scheme in the surf zone.

5. Laboratory experiments A limited set of laboratory experiments was conducted in a wave flume at the Universidad de Cantabria, to observe the existence of the mean flow prescribed in the analytical derivation proposed above. The 24 m long, 0.58 m wide and 0.80 m high wave flume has been provided with a series of slotted horizontal plates at the still water level, in order to absorb the wave transmitted by the structure without breaking. The rectangular breakwater, built of stones with D50 s 3.0 cm and ´ s 0.521, enclosed in a wire mesh was 0.76 m long, 0.50 m high and 0.58 m wide. Experiments were carried out with h s 0.35 m, T s 1.5 s, kh s 0.88 and varying incident wave height Ž0.04 m - H - 0.06 m.. The mean flow was observed using red dye introduced into the structure at different locations. In order to observe the influence of the location, three tubes where vertically introduced into the structure at different locations and varying depths ranging from two diameters from the still water level to 2 cm from the bottom of the structure. This kind of visualization requires diffusion to be small and therefore, it is necessary to work with small stone diameters. The experiments were carried out with and without structure in the wave flume in order to assure that the mean flow is induced by the presence of the structure only.

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Fig. 9. Stream lines of the return flow for a finite breakwater with a backwall. br hs1, khs 0.34 and ´ s 0.5.

Although no measurements were taken, the experiments showed a qualitatively good agreement with the patterns predicted by the theory developed. In all cases, independently of the location and depth of the tubes, the dye came out through the front breakwater side and usually close to the bottom proving the existence of the prescribed returning mean flow. Plate 1 shows one of the experiments for H s 0.04 m where the dye has been injected at the front tube located at mid-depth. The plates show the experiment at three different instants. It can be observed how the dye is coming out from the front face and mainly close to the bottom. This case has been selected since, due to the short distance between the tube location and the front face, the concentration of the dye is high and, therefore, easy to visualize.

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Fig. 10. Stream lines of the return flow for a semi-infinite breakwater. D50 s 3.0 cm, ´ s 0.521, bs 0.76 m, hs 0.35 m, T s1.5 s, khs 0.88, H s 0.04 m.

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Plate 1. Experimental verification of the mean return flow at a vertical permeable breakwater at three different instants.

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6. Conclusions In this paper, an analytical solution to the evolution of mean quantities and mean flows in permeable vertical structures is presented as a first approach to the problem, considered to be of considerable importance for the stability and the hydraulic performance of rubble mound breakwaters. It has been shown that second-order mean quantities, such as mass transport or mean water levels in the neighbourhood or inside breakwaters, are controlled by wave reflection and dissipation. Therefore, the different variables affecting wave reflection and dissipation Žincident wave conditions, structure geometry and material characteristics. do control the mean quantities patterns and magnitude. In general, the mean water level presents a modulation in front of the structure due to wave reflection. This modulation is stronger with increasing wave reflection. Within the structure, the mean water level is also modulated, increasing towards the lee of the structure. This set-up is induced by the reduction in wave height due to dissipation inside the structure. Modulation and magnitude are highly dependent on wave incidence characteristics, breakwater geometry and porous material characteristics. Furthermore, it has been shown that the rapid decrease in the Eulerian mass transport results in a mean vertical velocity able to induce a mean flow that exits mainly at the toe of the structure. The higher the reduction in mass flux the stronger the mean return current, which can be expressed analytically in terms of a mean stream function. A limited set of experiments have qualitatively shown the presence of the mean flow. Further research has to be carried out in order to quantify the importance of the flow and its consequences on structure toe stability as well as to extend the approach to other geometries. Acknowledgements I.J.L and M.A.L. acknowledge the funding provided by the Commission of the European Communities in the framework of the Marine Science and Technology Programme ŽMAST., under the No. MAS3-CT97-0081 project ‘Surf and swash zone mechanics’ ŽSASME.. Appendix A. First-order solution A.1. Solution for a semi-infinite width breakwater To provide a simple example of the wave-induced flow within a breakwater, the breakwater will have a vertical face at x s 0 and extend to infinity in the positive x direction. ŽIn fact, the breakwater needs only be wide enough for the wave motion to be damped out.. The waves will be incident from the yx direction. Within the breakwater, the wave action will monotonously decay in the x direction, eventually going to zero. The rate of decay is relatively rapid with x. DLM introduces the skin depth as a measure of the distance into the breakwater where the motion is significant. The skin depth is defined here as d s y1rK I , where K I is the Žnegative. imaginary part of the wave number in the porous region.

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The velocity potentials in each region that satisfy the Laplace equation and the associated boundary conditions are ŽDLM.:

f1 s f2 s

ai g cosh k Ž h q z .

v

w eyi q x q Re i q x x e i v t eyi l y

cosh kh

ai g

v

A

cosh K Ž h q z .

eyi ŽQ xy v t . eyi l y

cosh Kh

Ž A.1 . Ž A.2 .

where a is the incident wave amplitude, R is the a priori unknown reflection coefficient, resulting from the interaction of the incident wave train and the breakwater, A is the unknown amplitude of the wave motion in the porous medium and the oblique incidence is introduced in the solution via q s 'k 2 y l2 and Q s 'K 2 y l2 where l s ksin u . The angular frequency v is related to the water depth and wave number by the two dispersion relationships outside and inside the structure:

v 2 s gk tanh kh

Ž A.3 .

v 2 Ž s y i f . s gK tanh Kh

The wave number in the porous medium, K, is complex, with the Žnegative. imaginary part, K I , causing the waves to decay exponentially within the structure. To match the solution in the porous medium to the solution in the water region, the continuity of mass flux, which can be reduced to continuity of horizontal velocity across the vertical porous interface and the continuity of pressure is required at x s 0, yh F z F 0.

Ž1 qR.

cosh k Ž h q z . cosh kh

qŽ1 yR.

s Ž s y if . A

cosh k Ž h q z . cosh kh

s ´ QA

cosh K Ž h q z . cosh Kh

cosh K Ž h q z . cosh Kh

Ž A.4 . Ž A.5 .

These equations are functions of depth and cannot be solved for A and R directly. By multiplying both sides by cosh k Ž h q z .rcosh kh and integrating over depth, the vertical variation of the velocity and pressure associated with the progressive modes can be approximately determined.

Ž1 qR. G s Ž s yi f . Ax

Ž A.6 .

Ž 1 y R . q G s ´ AQ x

Ž A.7 .

where the following constants are defined:

Gs

0

xs

Hyh 0

Hyh

ž ž

cosh k Ž h q z . cosh kh cosh k Ž h q z cosh kh

2

/ . /ž

dz cosh K Ž h q z . cosh Kh

/

dz

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Solving this now depth-independent set of equations yields: Rs As

1ym 1qm

Ž A.8 .

2G

x Ž s y i f . Ž 1 q m.

.

Here, ms

´Q

Ž syi f . q

,

Ž A.9 .

the parameter m is the admittance of the structure ŽDLM., defined as the ratio of normal velocity at the breakwater wall to the pressure. A high value for the admittance means that the wave motion is easily transmitted into the structure. Using Eq. Ž9. and substituting f yields the linear displacement:

h Ž x ,t . s a Ž s y i f . Aeyi Q x e i v t eyi l y .

Ž A.10 .

A.2. Solution for a finite width breakwater If the breakwater width is small compared to the skin depth, a finite width breakwater must be considered. Then, the velocity potentials in each region simplify to:

f1 s f2 s f3 s

ai g cosh k Ž h q z .

v cosh kh ai g cosh K Ž h q z . v cosh Kh ai g cosh k Ž h q z . v

cosh kh

w eyi q x q Re i q x x e i v t eyi l y w Aeyi Q x q Be i Q Ž xyb . x e i v t eyi l y

Ž A.11 .

T eyi qŽ xyb. e i v t eyi l y

Following the same procedure as before, the following solutions are obtained: Rs Ts

i Ž 1 y m 2 . sin Ž Qb . 2 m cos Ž Qb . q i Ž 1 q m2 . sin Ž Qb . 2m

2 m cos Ž Qb . q i Ž 1 q m2 . sin Ž Qb . T G 1 As 1q 2 EŽ s y i f . x m T G 1 Bs 1y , m 2 EŽ s y i f . x

Ž A.12 .

where E s eyi Q b .

Ž A.13 .

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R and T are functions of only two parameters, m, the dimensionless admittance of the breakwater and bQ, a dimensionless width of the structure. The free surface elevation is now given by:

h Ž x ,t . s a Ž s y i f . Ž Aeyi Q x q Be i Q Ž xyb . . e i v t eyi l y .

Ž A.14 .

A.3. Solution for a finite breakwater backed by an impermeable wall For this case, the breakwater is limited in width by an impermeable wall at x s b. With the plane wave approximation, the velocity potentials are:

f1 s f2 s

ai g cosh k Ž h q z .

v cosh kh ai g cosh K Ž h q z .

v f 3 s 0.

cosh Kh

w eyi q x q Re i q x x e i v t eyi l y w Aeyi Q x q Be i Q Ž xyb . x e i v t eyi l y

Ž A.15 .

Substituting the potentials into the matching conditions and using the orthogonality relationships yields: Rs As

1 y i m tan Ž Qb . 1 q i m tan Ž Qb . 2 G

E

Ž A.16 .

Ž s y if . x cos Ž Kb . q i m sin Ž Kb .

B s AE 2 , where E s eyi Q b .

Ž A.17 .

The expressions for free surface elevation are the same given for the previous case but using the R, A, B given by Eq. ŽA.16..

Appendix B. Derivation of the mean stream function Here, the derivation of the mean stream function is presented. The stream function for the finite breakwater case is derived only since the semi-infinite and backwall cases are simplifications of the most general case. Consider the interaction of a normally incident wave train with a porous structure of width, b, between two semi-infinite fluid regions of constant depth, h. After solving the first-order problem, the boundary value problem for the mean stream function can be defined in every region in terms of the following governing equation:

= 2Ci s 0

½

j i F x F gi i s 1,2,3, yh F z F 0

Ž B.1 .

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with i s 1 for the water region seawards the structure, i s 2, for the porous region defined by the structure geometry and i s 3, the fluid region located leewards the structures. Therefore,

Ž j 1 ,g 1 . s Ž y`,0 . Ž j 2 ,g 2 . s Ž 0,b . Ž j 3 ,g 3 . s Ž b,` . Boundary conditions for the problem are given by the free surface and bottom boundary conditions. The free surface boundary conditions in each region is given by:

Ci x s V i , z s hi , i s 1,2,3

°V s 0 sw ¢V s 0,

where~V

1

Ž B.2 .

2 3

where w is the mean vertical velocity. The bottom boundary conditions can be expressed as:

Ci x s 0, z s yh, i s 1,2,3.

Ž B.3 .

Furthermore, a condition of finiteness in x has to be imposed to close the problem. Applying Taylor series at z s 0 to Eq. ŽB.2. and keeping the linear terms only results in:

Ci x s V i , z s 0,

Ž B.4 .

that should be used instead of Eq. ŽB.2.. The solution of the problem in every region is found by separation of variables. In region Ž1. and using the boundary conditions ŽB.4. and ŽB.3. for i s 1, as well as the requirement of the solution to be finite, the general expression of the solution is: ` np C 1 Ž x , z . s Ý Cn sin Ž h q z . e np x r h q c1Ž2. z q d1Ž2. . Ž B.5 . h ns1 Similarly, the general expression in the leeward region, region Ž3., can be obtained resulting in the following equation: ` np C 3 Ž x , z . s Ý Dn sin Ž h q z . eyn p Ž xyb.r h q c3Ž2. z q d 3Ž2. , Ž B.6 . h ns1 where the exponential variation in x has been referred to x s b. Finally, for the solution in region Ž2., applying separation of variables and the boundary condition ŽEq. ŽB.3.. for i s 2, the following general expression can be obtained: Ž2. C 2 Ž x , z . s Ž a2Ž 1 . e m R x q b 2Ž1. ey m R Ž xyb. . sin m R Ž h q z . q aŽ2. 2 Ž z q h . x q c2 z

q d 2Ž2. q Ž a2Ž 3 . e i m I Ž xyb r2. q b 2Ž3. eyi m I Ž xyb r2. . sinh m I Ž h q z . ,

Ž B.7 .

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where m R and m I are the eigenvalues that correspond to the different values of the separation variable m2 . The condition ŽEq. ŽB.4.. at the free surface has to be applied. Therefore, and using the first-order solution we get: ga2

Ž sK R y fK I . K I  AA) e 2 K I x q BB ) ey2 K I Ž xyb . 4 v q Ž sK I q fK R . K R  AB ) eyi K R Ž 2 xyb . q BA) e i K R Ž 2 xyb . 4 e K I b ,

w s C 2 x Ž x ,0 . s

Ž B.8 .

from where the eigenvalues can be obtained:

mR s 2 K I ,

Ž B.9 .

mI s 2 KR .

Ž B.10 .

Furthermore, five additional constants can be determined: aŽ2. 2 s 0, aŽ1. 2 s

1 a2 g 2 v

b Ž1. 2 sy

aŽ3. 2 sy

b Ž3. 2 s

Ž B.11 . AA)

1 a2 g 2 v i a2 g 2 v

i a2 g 2 v

Ž sK R y fK I . sin2 K I h

BB )

BA)

AB )

,

Ž B.12 .

Ž sK R y fK I . sin2 K I h

Ž sK I q fK R . sinh2 K R h

Ž sK I q fK R . sinh2 K R h

,

Ž B.13 .

eKIb,

Ž B.14 .

eKIb.

Ž B.15 .

Finally, the resulting expression for the stream function in region Ž2. is given by:

C2 Ž x , z . s

1 a2 g 2 v

Ž sK R y fK I .

sin2 K I Ž h q z . sin2 K I h

=  AA) e 2 K I x y BB ) ey2 K I Ž xyb . 4 1 a2 g q

2 v

i Ž sK I q fK R .

sinh2 K R Ž h q z . sinh2 K R h

=  AB ) eyi K R Ž 2 xyb . y BA) e i K R Ž 2 xyb . 4 e K I b q c2Ž2. z q d 2Ž2. .

Ž B.16 .

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Expressing the complex amplitudes A and B in terms of their real and imaginary components, Eq. ŽB.16. may be rewritten as:

C2 Ž x , z . s

1 a2 g 2 v

Ž sK R y fK I .

sin2 K I Ž h q z . sin2 K I h

=  < A < 2 e 2 K I x y < B < 2 ey2 K I Ž xyb . 4 a2 g q

v

Ž sK I q fK R . e K I b

sinh2 K R Ž h q z . sinh2 K R h

=  cos K R Ž 2 x y b . w A R B I y A I BR x q sin K R Ž 2 x y b . = w A R BR q A I B I x 4 q c 2Ž2. z q d 2Ž2. .

Ž B.17 .

where A s A R A I i and B s BR q BI i. Constants c2Ž2. and d 2Ž2. have to be determined using the matching conditions at the interface between regions. B.1. Mass flux balance One of the matching conditions that has to be satisfied is continuity of mass flux at both interfaces. In front of the structure, mass flux results from the interaction between incident and reflected waves and can be expressed in terms of the reflection coefficient, R: 1

a2 k

M1 s r g Ž 1 y RR ) . . 2 v

Ž B.18 .

Leewards of the structures the mass flux is a function of the transmitted wave and therefore may be written as: 1 ka2 M3 s r g TT ) . 2 v

Ž B.19 .

Therefore, the porous medium induces a reduction of the horizontal mass flux that may be evaluated as: 1 ka2 M1 y M 3 s r g Ž 1 y RR ) y TT ) . . 2 v

Ž B.20 .

Based on conservation of mass it can be shown that:

r

b

H0

EC 2 Ž x , z . Ex

´ d x s M1 y M 3 ,

Ž B.21 .

where ´ has been included to take into account that the mass flux takes place through the pores only.

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The mean return flow in region Ž1. emanating from the structure is given by Eq. ŽB.18. and therefore, the following equation has to be satisfied:

EC 1 Ž x , z .

0

Ž M1 y M3 . s rH

d z. Ez can be evaluated:

Ž B.22 .

yh

From where, c1Ž2. M1 y M 3 c1Ž2. s s U1 y U3 . rh

Ž B.23 .

Assuming U1 y U3 to be the mass flux out of the structure, there is no outgoing flux in the leeward region and therefore, c 3Ž2. s 0.

Ž B.24 .

Using the expression ŽB.16. for the stream function in region Ž2., it can be shown that:

´C 2 z Ž 0,0 . s U1 ,

Ž B.25 . Ž B.26 .

´C 2 x Ž b,0 . s U3 .

After some algebra, we have: U3 c 2Ž2. s y . Ž B.27 . ´ Finally, and taking the bottom, z s yh as the reference level for the streamlines, i.e., C ,C 1Ž x,y h. s C 2 Ž x,y h. s C 3 Ž x,y h. s 0, the new expressions are: ` np C 1 Ž x , z . s Ž U1 y U3 . Ž z q h . q Ý Cn sin Ž h q z . e np x r h , Ž B.28 . h ns1 1 a2 g

C2 Ž x , z . s

2 v

Ž sK R y fK I .

sin2 K I Ž h q z . sin2 K I h

2

a g q

v

Ž sK I q fK R . e K I b

 < A< 2 e2 K

I

x

y < B < 2 ey2 K I Ž xyb . 4

sinh2 K R Ž h q z . sinh2 K R h

=  cos K R Ž 2 x y b . w A R B I y A I BR x q sin K R Ž 2 x y b . yU3 = w A R BR q A I B I x 4 q Ž z q h. , ´ ` np C 3 Ž x , z . s Ý Dn sin Ž h q z . eyn p Ž xyb.r h . h ns1

Ž B.29 . Ž B.30 .

In order to determine the coefficients Cn and Dn , the mean horizontal velocities are assumed to be equal at the interfaces, i.e.,

EC 1

´

s´

Ez EC 2 Ez

s

EC 2 Ez EC 3 Ez

at x s 0,

Ž B.31 .

at x s b.

Ž B.32 .

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It can be shown that these conditions are equivalent in order to assume that at the interface the streamlines are equal.

C 1 s ´C 2 at x s 0,

Ž B.33 .

´C 2 s C 3 at x s b.

Ž B.34 .

Applying the orthogonality of the eigenfunctions  sin np Ž h q z .rh, n s 1,2, . . . 4 and  cos np Ž h q z .rh, n s 1,2,... 4 , the coefficients Cn and Dn : Cn s

´ a2 g Ž sK R y fK I . Ž AA) y BB ) e 2 K I b . 4 K I2 cos np v np q

4 K I2 y Ž nprh .

2

8 e K I b Ž sK I q fK R . K R2 cos np 4 K R2 q Ž nprh .

2

=  cos K R b w A R BI y A I BR x y sin K R b w A R BR q A I B I x 4 ,

Dn s

Ž B.35 .

´ a2 g Ž sK R y fK I . Ž AA) e 2 K I b y BB ) . 4 K I2 cos np v np q

4 K I2 y Ž nprh .

2

8 e K I b Ž sK I q fK R . K R2 cos np 4 K R2 q Ž nprh .

2

=  cos K R b w A R B I y A I BR x q sin K R b w A R BR q A I B I x 4 .

Ž B.36 .

B.1.1. Semi-infinite breakwater The solution to the stream function can be found from Eq. ŽB.29., simply by assuming B s 0, T s 0 and U3 s 0. The reflection coefficient R and the complex amplitude A have to be determined using the first-order solution or a semi-infinite breakwater in the appendix. ` np C 1 Ž x , z . s U1 Ž z q h . q Ý Cn sin Ž h q z . e np x r h Ž B.37 . h ns1

C2 Ž x , z . s Cn s

ga2 < A < 2 Ž sK R y fK I . sin2 K I Ž h q z . 2 v sin2 K I h

4´ a2 g < A < 2 Ž sK R y fK I . K I2 cos np

v np

4 K I2 y Ž nprh .

2

e2 K I x

Ž B.38 . Ž B.39 .

B.1.2. Finite breakwater backed by an impermeable wall The expressions for the stream function are the same given for the finite breakwater case, but using the R, A and B given by Eq. ŽA.16.. Furthermore, U3 is taken to be zero since there is no mass flux through the wall.

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