Exact and asymptotic analysis of waves generated by sea-floor disturbances on a sloping beach

Arghya Bandyopadhyay,Maria Victoria Otero-Espinar

doi:10.1186/s13661-015-0315-7

Abstract

A two-dimensional problem of small amplitude waves generated by some sea-bottom disturbance is studied on a sloping beach. The exact analytical solution for the wave height is provided, and with a periodic ground motion the asymptotic analysis of the waves is shown to exist for all time even in the vicinity of the shoreline. The novelty of this work lies in solving the corresponding Fredholm integral equation of the first kind and also to provide a uniform asymptotic estimate of the wave integral in the unsteady state involving both pole and saddle points for which Van der Waerden’s method is used. Within the framework of linear irrotational theory, explicit integral solutions of waves and their asymptotic and/or numerical computations presented here aim to provide an equivalent mathematical understanding to all such wave propagation which can be modelled in two dimensions at the open sea.

Highlights

The subject of waves on a fluid of variable depth has involved wide attention in a renewed way with the increased interest of researchers studying extreme waves, like tsunami, to improve the related knowledge in an attempt to reduce the gap between theory and application [ – ]
We are not dealing with tsunami waves as such, at the outset it is sensible to note that tsunami waves in the mid ocean are, small amplitude waves moving with tremendous velocity
7 Conclusion The problem that we have examined here is a potential problem on a beach where waves are generated by a disturbance at the sea-bed with very little restrictions on the nature of the disturbance except for the oscillation

Summary

Introduction

The subject of waves on a fluid of variable depth has involved wide attention in a renewed way with the increased interest of researchers studying extreme waves, like tsunami, to improve the related knowledge in an attempt to reduce the gap between theory and application [ – ]. Assuming an arbitrarily distributed ground motion oscillatory in time, the formal solution to the problem is subjected to an asymptotic analysis by Van der Waerden’s method [ ] which provides uniform asymptotic estimate of the wave height in an unsteady motion even in the vicinity of the critical line. The fact that most of the tsunami energy is transmitted at right angles to the major axis allows researchers to regard in this context the propagation of the tsunami in the open sea as being two-dimensional [ ] By this we surmise that mathematical analysis of similar nature may be adopted while analysing wave integrals for the tsunami wave propagation at the open sea where the wave amplitudes are small. The above expression ( . ) is the exact integral solution of the problem as formulated in ( . )-( . )

Detailed asymptotic analysis of η for a time-periodic ground motion

Displacement at the origin We consider the disturbance model given by

Conclusion