Surface Wave Scattering by an Elastic Plate Submerged in Water with Uneven Bottom

Wave interaction with a vertical elastic plate in presence of undulating bottom topography is considered, assuming linear theory and utilizing simple perturbation analysis. First order correction to the velocity potential corresponding to the problem of scattering by a vertical elastic plate submerged in a fluid with a uniform bottom is obtained by invoking the Green’s integral theorem in a suitable manner. With sinusoidal undulation at the bottom, the first-order transmission coefficient (T1) vanishes identically. Behaviour of the first order reflection coefficient (R1) depending on the plate length, ripple number, ripple amplitude and flexural rigidity of the plate is depicted graphically. Also, the resonant nature of the first order reflection is observed at a particular value of the ratio of surface wavelength to that of the bottom undulations. The net reflection coefficient due to the joint effect of the plate and the bottom undulation is also presented for different flexural rigidity of the plate. When the rigidity parameter is made sufficiently large, the results for R1 reduce to the known results for a surface piercing rigid plate in water with bottom undulation.

many researchers all over the world due to their importance in understanding the wave energy distribution near coastal regions. Some naturally occurring bottom obstacles are shore-parallel bars, sand ripples etc. which create obstructions to shoreline erosion by dissipating incident wave energy.
Although the exact forms/expressions of the reflection and transmission coefficients are difficult to obtain analytically, they can be estimated numerically. In relevant literature, we find that several mathematical techniques have been adopted to examine the effects of different bed-forms. One such method is based on the mapping of the actual fluid domain into some simpler ones, normally into a uniform strip (e.g. Kreisel [10], Roseau [25], Fitz-Gerald [7]). But such mappings are not convenient for arbitrary bottom variations. A numerical method based on integral equation formulation was applied by Newman [20], Miles [17] for simple topographic variation, such as step changes in depth. Davies [5] examined the problem of the interaction between surface waves and a patch of sinusoidal ripples in the bottom on the basis of linear perturbation theory followed by an application of Fourier transform technique. In his work, Davies [5] obtained the reflection and transmission coefficients for the 'far-field' and observed an oscillatory nature along with the resonance of the reflection. Later an extensive set of laboratory observations presented by Davies and Hearthershaw [6] confirmed the theoretical predictions made by Davies [5]. Staziker et al. [26] presented a mathematical method based on Green's function theory and a variational approach to study the problems of waves scattering by local bed elevation of arbitrary shapes.
There exists a class of problems dealing with small bottom deformations or slightly curved thin barriers which cannot be solved exactly but can be formulated by adding small correction terms to the mathematical solution of an exactly solvable problem. Perturbation analysis can be employed to deal with such problems. Perturbation analysis leads to an illustration in terms of a formal power series to get the required solution. We can see the use of perturbation technique in the work of several researchers. In the context of bottom undulations, Maiti and Mandal [12] investigated the scattering of incident waves in an ice-covered two-layer fluid. Banerjea et al. [1] used perturbation technique to study the effect of bottom undulation on the waves generated by an oscillating line source submerged beneath an ice-covered surface. Using regular perturbation analysis, Panda et al. [22] studied the flow problems in three layer fluid associated with small undulation at the bottom. Effect of the bottom undulations in a channel consisting of two immiscible fluids was considered by Mohapatra and Bora [19]. Later, Mohapatra [18] examined the effect of surface tension on the problem of wave scattering by bottom undulation in a two-layer fluid. Panda and Martha [21] considered a similar problem in presence of permeable bottom.
Apart from studying the sole effect of bottom undulation, it may be interesting as well as useful sometimes to investigate the joint effect of obstacles/surface discontinuity and bottom undulation. Since the bottom of an ocean is rarely of uniform depth throughout, it is worthwhile to study the effect of bottom undulation on wave-structure interaction problems for addressing realistic physical situations. Whenever uneven bottom is present, it causes wave transformation through the reflection of incoming waves and other phenomena like wave refraction and shoaling. Thus it is important to consider bottom undulations for modeling barriers. Again, the elastic plate has its own importance in constructing breakwaters due to longevity and cost-effectiveness. Also, the mathematical modelling becomes challenging when the propagating waves interact with the barrier as well as with the bottom undulations. These have motivated us to investigate the joint effect of the submerged vertical elastic plate and the bottom undulations. In literature, we can find some papers which study the combined effects of different types of obstacles and bottom undulations. Mandal and Gayen [14] employed perturbation analysis to investigate the problem of wave scattering by a surface-piercing barrier in presence of undulating bottom. Rakshit and Banerjea [24] used a similar perturbation technique to study the effect of small bottom undulation on the waves generated by rolling vertical plate either partially immersed or fully submerged in water. Again, Mandal and De [13] investigated the problem of wave scattering by bottom undulation in presence of a surface discontinuity. Choudhary and Martha [4] extended the work of Mandal and Gayen [14] for two different barrier configurations and included the graphs for hydrodynamic force and moment with respect to plate length.
In the context of linear water wave theory, wave interaction with submerged elastic plates is not a well studied topic. Meylan [16] investigated the problem of wave scattering by surface piercing vertical elastic plate to study the nature of the reflection and transmission. Peter and Meylan [23] used eigenfunction expansion to handle the time-dependent version of Meylan's [16] problem. Chakraborty et al. [2] presented a hypersingular integral equation technique to study the problem of wave scattering by a thin vertical elastic plate. Kundu et al. [11] used the method of Chakraborty et al. [2] to investigate the scattering problem in presence of an elastic plate with arbitrary inclinations.
In the present paper, we employ the aforesaid perturbation method to study the problem of wave scattering by a submerged vertical elastic plate in presence of undulating bottom of the domain. Application of the perturbation method produces boundary value problems (BVPs) for the potential functions of order zero and one respectively. The first problem (henceforth denoted as BVP-0) is related to the problem of wave scattering by a vertical elastic plate in uniform finite depth. The second boundary value problem (BVP-1) represents the radiation problem in water of uniform finite depth and involves the solution of BVP-0 in its bottom condition. Without solving BVP-1 explicitly, using Green's integral theorem the first order reflection and transmission coefficients can be represented in terms of the shape function (describing the bottom undulation) and the solution of BVP-0. The expression of the first order reflection coefficient (R 1 ) involves a particular form of the normal derivative of the zeroth order potential function. To find that we use a predefined mathematical technique based on the applications of the Green's function, Green's integral theorem and an expansion-collocation method (cf. Chakraborty et al. [2]). On the other hand, the first order transmission coefficient (T 1 ) vanishes identically with the assumption of symmetry of sinusoidal undulations at the bottom. Finally, |R 1 | is depicted graphically for several values of the flexural rigidity, barrier depth, ripples' amplitude and ripple numbers. From these results, and from the depicted values of the first-order corrections to the zeroth order reflection coefficient (R 0 + R 1 ) with respect to the dimensionless wave number, one can understand the joint effect of a vertical elastic plate and the undulating bottom topography on the propagation of surface waves. Graphical representation of |R 1 | with respect to the ratio of incident wavelength and the wavelength associated with bottom undulation is given. This gives a clear idea about the resonant nature of the first order reflection coefficient near a particular value of the above mentioned ratio. The results for |R 1 | are verified in two ways, one by comparing the known results for a surface piercing rigid plate to those for an elastic plate with sufficiently large rigidity; another by comparing the existing results for the sole effect of bottom undulations to those for a negligible length of the elastic plate.

Mathematical formulation
A two-dimensional potential flow is considered in the water domain of finite depth. In the corresponding Cartesian coordinate system y = 0 represents the undisturbed free surface and the positive y-axis is taken vertically downwards (see Figure 1). The bottom of the domain has a small undulation which is described by y = h + c(x). Here the continuous and bounded function c(x) indicates the shape of the undulation and is thus non-zero within the interval of x-axis where the undulation is present. Beyond this interval, c(x) is identically equal to zero. The dimensionless parameter ( 1) represents the smallness of the irregularity at the bottom of the water domain. The vertical elastic plate occupies the region x = 0, l 1 < y < l 2 . It is assumed that the motion is irrotational, time harmonic and the fluid is inviscid and incompressible. Again, considering the linear water wave theory the velocity potential can be expressed as Re{φ(x, y)e −iσt } (cf. [3]) where σ is the angular frequency. Then the complex valued potential function φ(x, y) satisfies with K = σ 2 /g (g is acceleration due to gravity), n denoting the normal direction at any point of the undulating bottom. We should also specify the edge condition r indicating the distance of an arbitrary point in the water region from either end of the elastic plate and the radiation conditions and k 0 is the real positive root of k tanh kh = K. In equation (2.5), R and T represent the reflection and transmission coefficients respectively.
The condition on the elastic plate is given by (cf. [16]) Here φ − = φ(0 − , y) and φ + = φ(0 + , y) denote the potentials on the left and the right sides of the plate. D = Eh 3 12ρ(1−ν 2 )g is the stiffness of the elastic plate, E represents the Young's modulus, ν denotes the Poisson's ratio. Again, m = ρ /ρh , ρ and ρ are the densities of water and plate material, h is its negligible thickness. Also, when the top end of the plate is clamped and the bottom end is free then the end conditions are given by

Method of solution
To implement the solution procedure as described in Section 1 we observe that the bottom condition (2.3) can be expressed as (cf. [14]) The form of the above approximation (3.1) and the smallness of the parameter indicate that the potential function and the constants R and T can be expanded as Substituting the above expansions in (2.1)-(2.2), (2.4)-(2.6), (3.1) and equating the coefficients of up to first order terms we find that φ 0 and φ 1 satisfy two boundary value problems namely BVP-0 for φ 0 and BVP-1 for φ 1 . BVP-0 (for the function φ 0 (x, y)): BVP-1 (for the function φ 1 (x, y)): The solution of BVP-1 depends on the solution of BVP-0. It is interesting to note that the values of R 1 and T 1 can be found without solving for φ 1 and simply using integrals involving c(x) and φ 0x as determined in equations (3.4) and (3.5). This can be achieved by applying Green's integral theorem on the potential functions φ 0 (x, y) and φ 1 (x, y) over the region bounded externally by the lines and internally by a contour enclosing the submerged elastic plate (see Figure 2).
Considering the limit X → ∞ and shrinking the contour enclosing the plate we determine R 1 as Again applying Green's integral theorem on φ 0 (−x, y) and φ 1 (x, y) in the same region we obtain Before we proceed further we must mention that our analysis is valid for small bottom deformations such as the presence of natural sand ripples at the bottom.
The ripples have similar forms as that of sinusoidal functions. We thus consider c(x) as Here M is a positive integer denoting the number of ripples at the bottom, λ is the wave number of each ripple and c 0 represents the ripple amplitude.
Substituting the above form of c(x) into equation (3.5) we find that T 1 vanishes identically where we have used the condition c(−x) = −c(x). Now, equation (3.4) can be manipulated as It may be noticed that the solution φ 0x to the differential equation (3.3) involves the unknown potential difference across the elastic plate. This unknown function is determined numerically by solving a hypersingular integral equation and using the solution, finally the normal derivative of φ 0 on the plate is calculated (cf. [11]). The method is briefly illustrated in Appendix.

Numerical results
In this section, we present some relevant graphs of absolute values of R 1 and R 0 + R 1 depending on the changes of different parametric values. Physical quantities are made dimensionless with respect to the uniform depth of the water domain. The plate is clamped at the upper end and free at the lower end.
In Figure 3, we present the values of |R 1 | as a function of Kh to compare the results for a rigid plate (cf. [14]) to that of the elastic plate. We consider a negligible submergence depth (l 1 /h = 0.001) of the upper end of the plate to compare our results with those for a partially immersed barrier. It is clear that when D is large enough [D/h 4 = 10], the absolute values of the first order reflection coefficient for an elastic plate completely agree with those for a rigid plate.         Figure 9 shows the effect of ripple numbers on |R 1 | for the fixed values of l 1 /h = 0.05, l 2 /h = 0.2 and c 0 /h = 0.1. The value of |R 1 | increases and becomes more oscillatory with increasing number of ripples. The reason behind this outcome is the multiple interactions of the incident wave with ripple tops. In Figure 10, the absolute value of the first order reflection coefficient is depicted against Kh for various ripple amplitude. It is clear that for higher ripple amplitudes |R 1 | increases. In this figure, the fixed values are l 1 /h = 0.01, l 2 /h = 0.2 and M = 1.
Bragg reflection over varying topography is an important topic to discuss in the context of problems dealing with bottom irregularities. Several researchers have studied the developed models of Bragg reflection (cf. Kirby [9]). Through these studies, ocean engineers can make a clear idea about the peak value of reflection due to bottom undulation and the condition of its occurrence. In early studies, it has been demonstrated that in a simple case of a patch of sinusoidal ripples in the bottom, the resonant nature of the first order reflection depends on the wavelengths of the incoming wave and that of the undulating bottom.  In Figures 11 and 12 we represent the graphs of |R 1 | with respect to α(= 2k0 λ ) for various values of L/h with fixed values of l 1 = 0.01, M = 2, c 0 /h = 0.1 and D/h 4 = 1. In Figure 11, Kh = 0.25 whereas in Figure 12, Kh = 1.5. The value of α for which |R 1 | attains its highest value is almost unity. Though in Figure 11, the differences between the peak values of |R 1 | for different values of L/h are not clearly visible but for a moderate value of Kh(= 1.5) it is detectable that with increasing plate length the peak values of |R 1 | are shifted to the left.

Conclusions
The problem of wave scattering by a thin elastic plate in the presence of an undulating bottom has been studied. Employing a simplified perturbation analysis the first-order reflection and transmission coefficients are expressed in terms of integrals involving the shape function describing the bottom deformation and the normal derivative of potential function related to the problem of uniform finite depth. A particular case of bottom deformation has been considered in the form of sinusoidal ripples. For this particular choice, the first order transmission coefficient vanishes and the first order reflection coefficient is calculated numerically and presented graphically for different parametric values. Due to multiple interactions of waves with an elastic plate and ripples at the bottom, |R 1 | shows an oscillatory nature. From the graphs presented above, it is quite clear that the value of |R 1 | increases in absence of a barrier (see Figure 7). It has been established through graphical comparison with published results (see Figure 1) that the values of |R 1 | in the presence of an elastic plate for large rigidity are the same as those in the presence of a rigid plate (cf. Mandal and Gayen [14]). The variations of |R 1 | and |R 0 + R 1 | with varying flexural rigidity of the plate show that the net reflection increases as the plate tends to become rigid. Comparison of graphs for |R 0 + R 1 | with those for |R 1 | clarify the fact that the joint effect of the plate and the small bottom undulation increases the reflection due to the sole effect of bottom undulation to some extent. |R 1 | has also been presented with respect to the parametric value α denoting the ratio of the wavelength of the incoming wave to that of the bottom undulation, from which an observation can be made about the resonant interactions of waves with the ripples. These observations might be useful in modelling a flexible breakwater placed in a fluid region with small bottom undulations.

Now the solution of the equation (3.3) is expressed as
where φ 01 ≡ φ − 0 andφ 02 ≡ φ + 0 . Note that the above expression is not enough for numerical computation of φ 0x as its right hand side involves the unknown potential difference function (φ 01 (η) − φ 02 (η)). We thus seek for an alternative expression of φ 0x (0, y). Applying Green's integral theorem to φ 0x (0, y) − e ik0x ψ 0 (y) and G(x, y; ξ, η) (the fundamental potential function due to line source at (ξ, η)) in the domain given in Figure 13 we obtain an integral representation of φ 0 (x, y) as (cf. Chakraborty et al. [2]) Differentiating both sides with respect to x and taking the limit x → 0 we get The integration is performed in the sense of Hadamard finite-part integral and thus a cross is put on the integral sign. The two expressions of φ 0x as given in 2k 2 n cos k n (h−η) cos k n (h−y) 2k n h + sin 2k n h .