WAVE REFLECTION AND TRANSMISSION OVER A SUBMERGED HORIZONTAL PLATE AND THROUGH A VERTICAL POROUS WALL

H. M. Hu

WEST Consultants, Inc.

11848 Bernardo Plaza Court, Suite-140B, San Diego, CA 92128, USA;

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Abstract: A two-dimensional analytical solution is presented to study the reflection and transmission of small-amplitude water waves propagating past a submerged horizontal plate and through a vertically-slitted wall. The velocity potential in each fluid domain is formulated using three sets of orthogonal eigenfunctions and the unknown coefficients are determined from the matching conditions between domains. The hydrodynamic force acting on the porous wall is then examined. The performance of the breakwater system is evaluated in terms of reflection and transmission coefficients. Several comparisons are conducted to test the analytical solution against available laboratory measurements. It is found that the wave reflection and transmission coefficients and the hydrodynamic force from the analytical solution are, generally, in good agreement with the measured data. It is shown that the submerged plate plays an important role in reducing the transmitted waves, especially at the longer wavelengths.

Keywords: wave reflection and transmission; submerged plate; porous wall; linear wave theory; wave force

1    Introduction

There are widespread applications of porous structures in coastal engineering. A porous structure allows the partial transmission of water waves with energy dissipation. Therefore, both the transmitted and the reflected wave heights as well as the resultant force acting on the porous structure can be reduced. However, such permeable structures are relatively ineffective in dealing with long waves. Since wave interaction with a submerged horizontal plate results in a shortening of the wave length on its shoreward side (Kojima et al., 1990), it might thus be anticipated that a porous structure located on the lee side of a submerged horizontal plate would effectively intercept long waves once their lengths have been shortened.

Many theoretical and experimental studies have been performed to analyze either wave motion over submerged structures or water wave transmission through a porous medium. Stoker (1957) presented results for the reflection and transmission of long waves by a plate which is fixed on the free surface. Analytical expressions for the reflection and transmission coefficients for long waves propagating over a submerged plate were obtained by Siew and Hurley (1977). Patarapanich  (1984) carried out a set of laboratory experiments to investigate the variation of the wave reflection with the plate length. Liu and Iskandarani (1989) applied the method of eigenfuction expansions to derive a semi-analytical solution for wave propagating over a submerged plate. General numerical methods, such as the boundary integral method developed by Liu and Iskandarani (1989) and the finite element method by Patarapanich and Cheong (1989), have also been used for modeling the wave propagation.

The problem of the reflection and transmission of small amplitude waves by a vertical porous breakwater has also been studied by Chwang and Dong (1984) and Wang and Ren (1993). In their studies, Taylor’s (1956) concept of a porous screen is used to avoid the complicated porous medium flow. Later, Ren and Wang (1994) extended their analytical analysis to investigate the behavior of a flexible, porous and floating breakwater system.

In this study, an analytical model based on the linear potential wave theory is developed to investigate waves propagating past a submerged horizontal plate and through a vertically-slitted wall.

2    Theoretical Formulation

The geometry of the problem is shown in Fig. 1. A train of small amplitude waves of height H, frequency w and wavelength  propagates in the positive x-direction and are normally incident upon a breakwater system consisting of a submerged horizontal plate and a vertically-slitted wall behind the plate. Cartesian coordinates are employed to define this two-dimensional boundary-value problem. The z axis is directed vertically upwards from an origin on the still water level in water of uniform depth . The plane z = 0 coincides with the undisturbed free surface. The plane x = 0 passes through the centerline of the plate. The length of the plate is equal to B. The horizontal plate is fixed at a certain depth h₁ below the quiescent water surface and considered as a thin structure. The distance between the plate center and the wall front along the wave direction is denoted by x_w. The vertical porous wall is also considered thin.

As shown in Fig. 1, the fluid domain is divided into five regions, which are separated by the upstream side of the front edge of the plate, the plate itself, the lee end of the plate, and the porous wall. The fluid is assumed to be incompressible and inviscid and its small-amplitude motion is irrotational. The fluid motion may then be described in terms of velocity potentials. The basic equation describing the fluid motion in the five regions is the Laplace equation,

                                (1)

where f_j denotes the velocity potential in the jth region ( j = 1, 2, 3, 4, 5). The combined linearized free surface boundary condition is given as

     at z = 0    ( j = 1, 2, 4, 5)               (2)

where g is the gravitational acceleration. The bottom boundary conditions are

    at z = -h                          (3)

for j = 1, 3, 4, 5. The kinematic  boundary condition on the plate surface requires that the normal velocity is equal to zero, that is,

    at z = -h₁,    -a £ x £ a                    (4)

    at z = -h₁,    -a £ x £ a                    (5)

By assuming that the velocity inside the porous medium depends only on vertical elevation, the boundary condition on the vertical porous wall surface can be expressed as

    at x = x_w     ( j = 4, 5)                 (6)

where W(z, t) is the normal velocity of the fluid passing through the porous wall from region 4 to region 5.

Matching conditions, representing continuity of pressure and velocity across the imaginary interface between the fluid regions, are imposed as

    ( j = 2, 3)     at x = -a                         (7)

     ( j = 2, 3)     at x = -a                     (8)

     ( j = 2, 3)     at x = a                         (9)

     ( j = 2, 3)     at x = a                    (10)

At large distances from the breakwater system, the velocity potential must satisfy the usual radiation condition, namely

                         (11)

where k₀ is the propagating wave number which satisfies the usual dispersion relation, .  The incident wave potential, f_I, is given by

                   ( 12)

The flow passing through the porous wall is assumed to obey Darcy’s law. Hence, the porous flow velocity W(z, t) is linearly proportional to the pressure difference between the two sides of the porous wall (Taylor, 1956), that is 

    at x = x_w                    (13)

where m is the constant coefficient of dynamic viscosity and b is a material constant having the dimension of a length. The hydrodynamic pressures p_j on the porous wall ( j = 4, 5) are related to the velocity potentials through the linearized Bernoulli equation

                            (14)

where r is the fluid density. From Eqs. (13) and (14), the expression for W(z, t) can be obtained as

    at x = x_w                    (15)

3    Analytical Solutions

The velocity potential in each fluid domain, satisfying the respective boundary conditions, can be obtained by the method of separation of variables. The unknown constants contained in the velocity potentials are determined by applying the matching conditions Eqs. (7)-(10) and further utilizing the orthogonal properties of three sets of eigenfunctions.

Once the velocity potential in each fluid domain is determined, various quantities of engineering interest, such as water surface elevations and the total net hydrodynamic force acting on the vertical permeable wall, may then be evaluated. The effectiveness of the submerged plate and vertical porous wall combination as a wave barrier can be measured in terms of a reflection coefficient, , and a transmission coefficient, , defined as

                                                             (16)

For , wave energy is conserved within the fluid system, otherwise, if , wave energy is dissipated by the porous medium.

4    Results and discussion

For the limiting case of a completely permeable vertical wall, the breakwater system is equivalent to a submerged horizontal plate. Comparisons of the analytical solutions with experimental measurements (Patarapanich and Cheong, 1989) for this limiting case are presented in Fig. 2 where the reflection coefficient is plotted against the ratio of plate length to the wave length in region 2, L₂. In general, the agreement between the analytical solutions and measured data is good. The present analytical solution tends to slightly over-predict the reflection coefficient in the case when the wavelength gives maximum reflection. This discrepancy is thought to be due to energy dissipation as the waves propagate past the horizontal plate. The effect of energy dissipation at the plate is not included in the present solution.

To examine the performance of the dual breakwater system consisting of a horizontal plate and a vertical porous wall, the variation of the reflection and transmission coefficients with the horizontal dimensionless distance, x_w/L, are presented in Fig. 3. In order to compare with the laboratory measurements, the values h/L = 0.15, h₁/h = 0.25, B/h = 2.0, and G₀ = 0.45 are considered. Here, G₀ is a dimensionless porous-effect parameter (Chwang and Li, 1983),  which depends on the fluid properties, the property of the porous wall, and the given incident wave. When G₀ = 0 the vertical wall is impermeable, while as G₀ approaches infinity, the porous breakwater becomes completely permeable. The measurements of C_r and C_t shown in Fig. 3 were obtained by Kojima et al. (1990). It can be seen that both the analytical solutions and the measured data indicate that with an increase of the distance ratio x_w/L, the transmission coefficient C_t and the reflection coefficient C_r vary periodically. The variation of the transmission coefficient obtained from the analytical solutions agrees reasonably well with the experimental measurements. However, the analytical solutions tend to over-predict the reflection coefficient in comparison with the measured data, which is again due to the unconsidered energy dissipation at the horizontal plate.

Figure 4 shows the dimensionless hydrodynamic force acting on the porous wall for the case of h/L = 0.15, h₁/h = 0.25, B/h = 2.0, and G₀ = 0.45. In Fig. 4, F_x is the total net hydrodynamic force on the porous wall. Clearly, the hydrodynamic force can be reasonably predicted by the present analytical model. The experimental values are due to Kojima et al. (1990). From Figs. 3 and 4, it is noted that as the distance ratio x_w/L increases, the wave force and the transmission coefficient vary in a similar manner.

The effect of incident wave characteristics on the breakwater performance is also studied. Figure 5 shows the variation of the transmission coefficient with the ratio of the water depth to the incident wave length, h/L. To show the influence of the horizontal plate, the transmission coefficients for the case without a plate are also presented in Fig. 9. The dimensionless plate width and submergence are B/h = 2.0 and h₁/h = 0.25. It can be seen that the transmission coefficients from the analytical solutions are in good agreement with the measurements (Kojima et al., 1990) at the longer wave lengths. Both the analytical solutions and the measured data indicate that the transmission coefficient tends to decrease initially with an increase in the relative water depth until it reaches a minimum value at h/L = 0.15, approximately. The transmission coefficient then increases as h/L increases. When h/L is greater than about 0.25, the transmission coefficient tends to a nearly constant value. It is interesting to note that the transmission coefficient for the present breakwater system (with a horizontal plate) is smaller than that when a single porous wall (i.e., without a horizontal plate) is used. The results suggest that a combination of a submerged horizontal plate and a porous wall can serve as an effective wave blocking device at longer wave lengths.

5    Conclusions

In this study, wave propagating over a submerged plate and through a vertical porous wall is investigated analytically based on linear wave theory. The velocity potentials in each fluid domain are derived and their unknown coefficients are determined from the matching conditions using three sets of orthogonal eigenfunctions. The performance of the breakwater system is evaluated in terms of the reflection and transmission coefficients. It is found that these coefficients and the hydrodynamic force acting on the porous wall vary periodically with an increase in the dimensionless distance between the plate center and the wall front. The analytical solutions agree reasonably well with published experimental data. It is also found that the variation of the transmission coefficient with the ratio of water depth to the incident wavelength is also in good agreement with published experimental measurements. The calculated changes in the wave transmission coefficient suggest that the present breakwater system can effectively reduce the transmitted waves, especially at longer wavelengths.

Acknowledgements

The author is grateful to Dr. K.-H. Wang and Dr. A. Neil Williams for their guidance and valuable review comments. 

References

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Fig. 1    Definition sketch of the breakwater problem.