Hydroelastic lumps in shallow water

TL;DR

This paper investigates hydroelastic solitary waves on elastic sheets over shallow water, deriving a new equation, analyzing wave types, and exploring stability and wave interactions through numerical simulations.

Contribution

It introduces a Benney-Luke-type equation for hydroelastic waves and analyzes the existence, stability, and interactions of various solitary wave solutions.

Findings

Identification of three solitary wave types: plane, lump, and transversally periodic

Discovery of a dimension-breaking bifurcation linking different wave solutions

Observation of lump shedding in response to moving localized loads

Abstract

Hydroelastic solitary waves propagating on the surface of a three-dimensional ideal fluid through the deformation of an elastic sheet are studied. The problem is investigated based on a Benney-Luke-type equation derived via an explicit non-local formulation of the classic water wave problem. The normal form analysis is carried out for the newly developed equation, which results in the Benney-Roskes-Davey-Stewartson (BRDS) system governing the coupled evolution of the envelope of a carrier wave and the wave-induced mean flow. Numerical results show three types of free solitary waves in the Benney-Luke-type equation: plane solitary wave, lump (i.e., fully localized traveling waves in three dimensions), and transversally periodic solitary wave, and they are counterparts of the BRDS solutions. They are linked together by a dimension-breaking bifurcation where plane solitary waves and lumps can be viewed as two limiting cases, and transversally periodic solitary waves serve as intermediate states. The stability and interaction of solitary waves are investigated via a numerical time integration of the Benney-Luke-type equation. For a localized load moving on the elastic sheet with a constant speed, it is found that there exists a transcritical regime of forcing speed for which there are no steady solutions. Instead, periodic shedding of lumps can be observed if the forcing moves at speed in this range.