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

Abstract

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.