Investigation of water wave scattering by an elastic sea-bed of varying depth in two superposed fluids covered by an ice-sheet

Abstract

A hydroelastic model is taken up in this work to investigate the propagation of normal incident waves over a small undulation located on an elastic sea-bed in a fluid flow for two superposed fluids where a floating ice-sheet covers the upper fluid. Here, both the floating ice-sheet and elastic sea-bed are approximated as thin elastic plates which follow the Euler–Bernoulli beam equation, familiar for such studies. Due to its minimal contribution, the surface tension at the interface of the fluid layers is not accounted for. The behavior of the roots of the associated complex dispersion relation is investigated through the application of a counting argument along with a contour plot. The time-harmonic waves, for a given frequency, are observed to propagate with exactly two wavenumbers such that the one corresponding to the smaller wavenumber propagates along the ice-sheet while the other corresponding to the higher wavenumber along the interface. By assuming the fluid to be incompressible and inviscid following an irrotational motion, and then applying a perturbation technique, the first-order contribution to the velocity potentials in each layer is evaluated with the help of Fourier transform. Following that, the associated reflection and transmission coefficients are computed by carrying out appropriate integration for integrals containing a shape function depicting the sea-bed undulation. In order to validate the present model developed in this work, one obtained result for reflection coefficient is validated against an available result which points towards a good agreement. This validation establishes the model and the method followed in this work to be effective and allows subsequent investigation for this problem to be taken up. For verification of the theory developed here, one specific undulating bed topography is selected to assess the reflection as well as transmission coefficients which are depicted by graphical representations to examine and establish the energy conversion between the flexural wave mode and the interfacial wave mode. Present investigation demonstrates that the reflection and transmission of the waves exhibit an oscillatory behavior, the reason for which is the outcome of multiple interaction of the incident wave with the elastic bottom undulation, interface and the ice-sheet. It further shows that flexural wave moment is significantly affected as compared to the interfacial wave influence. It is noted that significant variation in the elastic specification of the ice-sheet and the elastic sea-bed has a considerable influence for wave scattering by the undulating elastic sea-bed. It is observed that, when water waves propagate over a deformable bottom, the wave characteristics undergo a significant change. This occurs due to the elasticity of the sea-bed which immensely influences the kinematics and dynamics of the wave over and above providing the demonstration of the elastic behavior of the soil located below the elastic sea-bed. The approach adopted here is likely to be of immense significance in the design and construction of various types of floating breakwaters and very large floating structures (VLFS) for the purpose of various scientific activities mainly in polar regions. Further, the solutions of problems like these dealing with a deformable sea-bed will be beneficial in providing the required background to the researchers with respect to the incorporation of the characteristics of the infinite depth soil located below the sea-bed.