Abstract
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.
Highlights
Waves traveling over a seabed with uniform finite depth experience no reflection when there is no obstacle
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 φ0 and φ1 satisfy two boundary value problems namely BVP-0 for φ0 and BVP-1 for φ1
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
Summary
Waves traveling over a seabed with uniform finite depth experience no reflection when there is no obstacle. Mandal and Gayen [14] employed perturbation analysis to investigate the problem of wave scattering by a surface-piercing barrier in presence of undulating bottom. Meylan [16] investigated the problem of wave scattering by surface piercing vertical elastic plate to study the nature of the reflection and transmission. 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. 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 (R0 + R1) 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. The results for |R1| 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
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