Wave resonant scattering mechanism of sinusoidal seabed elucidated by Mathieu Instability theorem

Abstract

In this article, we shed light on unresolved issues on the physics of the problem, and clarify the effects of slope and other related parameters controlling the wave scattering phase shifting by sinusoidal seabed based on Mathieu Instability theorem. As a starting point, 3D water wave scattering by arbitrarily-varying sea bottom is governed by 2D wave potential and wavenumber coupling equation using Fredholm's Alternative Theorem. Seabed undulating effects are split into different components, including wavenumber and bottom slope and curvature effects. Sinusoidal ripples are used to demonstrate these different seabed undulating effects. The presented model predictions agree very well with lab data. Furthermore, a method with quantitative formulations is constructed and proposed to elucidate the resonance mechanism of wave scattering by sinusoidal seabed using Mathieu instability theorem. General stability chart is generated in a relevant parametric plane to detect the unstable conditions. Quantitively formulations of resonance conditions are presented to predict peak shifting features that beyond the Bragg's perdition. As demonstrated in this work, slope effect is contributed to peak shifting and the calculation formula of the shifting distance is given, though the shifting in single sinusoidal case is small, it is more obvious in double sinusoidal cases.