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.

Propagation of Oblique Waves Over a Small Undulating Elastic Bottom Topography in a Two-Layer Fluid Flowing Through a Channel

A hydrodynamic model, with the incorporation of elasticity, is considered to study oblique incident waves propagating over a small undulation on an elastic bed in a two-layer fluid, where the upper layer fluid is bounded above by a rigid lid, which is an approximation to the free surface. Following the Euler–Bernoulli beam equation, the elastic bed is approximated as a thin elastic plate. The surface tension at the interface of the layers is completely ignored since its contribution will be minimal. The behavior and properties of the roots of the dispersion relation are analyzed using counting argument and contour plot. In this case, time-harmonic waves propagate with only one wavenumber. Considering an irrotational motion in an incompressible and inviscid fluid, and applying perturbation technique, the first-order corrections to the velocity potentials are evaluated by an appropriate application of Fourier transform and, subsequently, the corresponding reflection and transmission coefficients are computed through integrals containing a shape function which depicts the bottom undulation. To validate the theory developed, two particular undulating bottom topography are taken up as examples in order to evaluate the hydrodynamic coefficients which are represented through graphs. It is noted that the reflection coefficient shows an oscillatory pattern, when the wavenumber of the undulating bed takes a value almost double the wavenumber of the incident wave. When the ratio of the wavenumber of the undulating bed and the wavenumber of the incident wave approaches 2, the theory predicts existence of resonance between the undulating elastic bed and the interface of the layers. It is worthwhile to note that reflected wave energy of the incident wave is higher if the number of ripples of the bottom undulation is large. It is further noted that the reflected energy increases in response to an increase in the values of the elastic parameters of the ocean bed. The results for a patch of sinusoidal ripples having different wavenumbers are found to be similar. Further, the reflected energy decreases for an increase in the angle of incidence. It is noted that reasonable changes in elasticity, ripple wavenumber and the number of the ripples of the bed have a remarkable effect when the propagating wave passing through a channel encounters a small bottom undulation. When results are compared with the available results, good agreement is observed.