Geometric Optics for Surface Waves in Nonlinear Elasticity

Abstract

This work is devoted to analysis of high frequency solutions to equations of nonlinear elasticity in a half-space. We consider surface waves (or more precisely, Rayleigh waves) arising in general class of isotropic hyperelastic models, which includes in particular Saint Venant-Kirchhoff system. Work has been done by a number of authors since 1980s on formulation and well-posedness of a nonlinear evolution equation whose (exact) solution gives leading term of an \emph{approximate} Rayleigh wave solution to underlying elasticity equations. This evolution which we refer to as the amplitude equation, is an integrodifferential equation of nonlocal Burgers type. We begin by reviewing and providing some extensions of theory of amplitude equation. The remainder of paper is devoted to a rigorous proof in 2D that exact, highly oscillatory, Rayleigh wave solutions $u^\eps$ to nonlinear elasticity equations exist on a fixed time interval independent of wavelength $\eps$, and that approximate Rayleigh wave solution provided by analysis of amplitude equation is indeed close in a precise sense to $u^\eps$ on a time interval independent of $\eps$. The paper focuses mainly on case of Rayleigh waves that are \emph{pulses}, which have profiles with continuous Fourier spectrum, but our method applies equally well to case of wavetrains, whose Fourier spectrum is discrete.