The influence of flexible bottom on wave generation by an oscillatory disturbance in the presence of surface tension

The two-dimensional problem of wave generation by a time-harmonic pressure distribution on the surface in a finite-depth ocean is studied in this article. Here, it is considered that the ocean has a flexible base, which is modelled as a thin elastic plate and is governed by the Euler-Bernoulli beam equation. The effect of surface tension at the free surface is also taken into account. Within the framework of the linearised theory of water waves, the initial boundary value problem is solved using the Laplace-Fourier transform technique and the integral form of the free surface elevation is obtained. The method of stationary phase is used to evaluate the asymptotic solutions of the free surface elevation for large time and distance. Different forms of the surface elevation have been demonstrated graphically for variation of parameters in a number of figures, and appropriate conclusions are drawn. Moreover, the dispersion relation associated with the wave motion is also derived and analysed using contour plots to understand the characteristics of the roots. Further, the phase and group velocities of the wave motion have also been investigated in this study for both shallow and deep water.