Nonlinear Surface Acoustic Waves and Waves on Stratified Media
Sections 1 and 2 proceed from Rayleigh wave theory for isotropic elasticity to an overview of linear and nonlinear surface waves on uniform anisotropic elastic and electro-elastic half-spaces. Section 3 concerns layering effects—in particular the existenc
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D.F. Parker University of Edinburgh, Edinburgh, UK
ABSTRACT Sections 1 and 2 proceed from Rayleigh wave theory for isotropic elasticity to an overview of linear and nonlinear surface waves on uniform anisotropic elastic and electro-elastic half-spaces. Section 3 concerns layering effects-in particular the existence of a shear-horizontal (Love) mode and the dispersion of the generalized Rayleigh mode. Section 4 outlines treatments of waves travelling along a wedge tip and of surface waves influenced by corrugation of the traction-free surface. Section 5 reformulates
some of the previously-derived nonlinear evolution equations in terms of a physically
relevant surface displacement, rather than its Fourier transform. This reveals the essentially non-local nature of the nonlinear evolution equations for surface waves, so distinguishing them from many others treated in this volume. CONTENTS 1. Nonlinear evolution of Rayleigh waves 2. Electro-elastic coupling and beam spreading 3. Nonlinear waves on layered materials 4. Waveguiding by wedges and surface corrugation 5. Nonlocal evolution equations References
A. Jeffrey et al. (eds.), Nonlinear Waves in Solids © Springer-Verlag Wien 1994
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D.F. Parker
1 NONLINEAR EVOLUTION OF RAYLEIGH WAVES 1.1
Introduction Elasticity is essentially a three-dimensional subject, so that many wave phenomena - reflection, refraction, scattering from inhomogeneities and from curved boundaries - are geometrically complex. For these, linear theory is usually adequate since, unless the effects of weak nonlinearity remain coherent, they cannot accumulate sufficiently to cause large response. For uni-directional problems geometric complexity is absent, disturbances travel essentially along chracteristics of a hyperbolic system and nonl inear evolution equations incorporating additional effects are readily derived. Waveguide problems also involve propagation essentially in one direction, but are much more intricate since they involve complicated distributions of stress and strain. Nevertheless, as a disturbance propagates, these complicated disturbations are essentially maintained by the guiding structure. The magni tude and distribution of nonl inear effects can then be related to the amplitudes associated with the relevant waveguide modes, so that evolution equations for scalar quantities may be derived for essentially one-dimensional propagation. The cases of wedge waves and waves in layered media have many features in common with those of elastic surface waves, so that this first chapter is used to describe the linear and nonlinear theories of such waves. 1.2
Linear-elastic surface waves We consider waves travelling in the positive Xl-direction along the surface X2 = 0 of an elastic half-space X2 < O. Here, XL (L = 1,2,3) are cartesian material (Lagrangian) coordinates. When a material point X moves to x = X(X,t) , with cartesian coordinates xJ (j = 1,2,3) , the velocity v and deformation gradient F have components F' L
J
= ox.loX L = x·J, L J
When thermal effects are neglected, t
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