Bifurcation and Secondary Bifurcation of Heavy Periodic Hydroelastic Travelling Waves

TL;DR

This paper investigates the bifurcation phenomena of steady hydroelastic travelling waves on deep ocean surfaces with elastic membranes, revealing a symmetry-breaking solution branch analogous to Wilton ripples.

Contribution

It introduces a bifurcation analysis of hydroelastic waves with a hyperelastic membrane, including the discovery of a symmetry-breaking solution branch.

Findings

Existence of bifurcation points for steady hydroelastic waves.

Identification of a symmetry-breaking 'third sheet' of solutions.

Analogy to Wilton ripples in hydroelastic context.

Abstract

The paper deals with a problem of interaction between hydrodynamics and mechanics of nonlinear elastic bodies. The existence question for two-dimensional symmetric steady waves travelling on the surface of a deep ocean beneath a heavy elastic membrane is analyzed as a problem in bifurcation theory. The behaviour of the two-dimensional cross-section of the membrane is modelled as a thin (unshearable), heavy, hyperelastic Cosserat rod, following Antman's elasticity theory, and the fluid beneath is supposed to be in steady 2D irrotational motion under gravity. Assuming that gravity and the density of the undeformed membrane are prescribed, the free parameters of the problem are the speed of the wave and drift velocity of the membrane. The analysis relies upon a conformal formulation of the hydro-elastic problem developed in previous papers; the basic tool for the study of the bifurcation picture is the implicit function theorem, under some non-resonance assumptions. The most interesting part of the final result is the existence of a symmetry-breaking 'third sheet' of solutions, which bifurcates from primary sheets, and is a hydro-elastic analogue of the phenomenon known as 'Wilton ripples' in the surface tension case.