The Quarterly Journal of Mechanics and Applied Mathematics Advance Access originally published online on October 31, 2008
The Quarterly Journal of Mechanics and Applied Mathematics 2009 62(1):19-30; doi:10.1093/qjmam/hbn022
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Rayleigh waves having generalised lateral dependence
( School of Mathematics, University of Edinburgh, Edinburgh EH9 3JZ )
( St. Petersburg Branch, Steklov Institute of Mathematics, St. Petersburg, 191023 Russia )
d.f.parker{at}ed.ac.uk
Received 22 April 2008.
Revise 30 September 2008.
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Abstract |
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It is shown that the surface-guided elastic waves found by Kiselev for isotropic materials and having displacements depending linearly upon the Cartesian coordinate orthogonal to the sagittal plane may be generalised in many ways. For surface waves on any anisotropic half-space, a simple procedure applied to the displacements within the standard surface wave having dependence ei
, where 
k · x – 
t and k is the (surface) wave vector, yields displacements depending linearly upon the surface cartesian coordinate orthogonal to the group velocity vector. Moreover, repeated application of this (differentiation) procedure yields a hierarchy of waves having algebraic dependence of successively increasing degree. For isotropic materials, substantial simplification and generalization are possible. Solutions of all algebraic degrees have identical depth dependence. This allows the solutions to be constructed iteratively and motivates a search for general solutions having depth dependence of the normal displacement the same as in the standard surface wave. The procedure gives a new derivation of the solutions found by Achenbach having amplitude of the normal displacement of the surface given by any solution to the two-dimensional Helmholtz equation. Furthermore, exploiting the scale invariance (a property of surface waves on any homogeneous half-space) shows that in every surface-guided disturbance of an elastic half-space, the elevation of the free surface is a solution of the wave equation in two dimensions (the membrane equation). Using the paraxial approximation to the membrane equation, high-frequency Rayleigh waves propagating as narrow beams are described in terms of a scalar Gaussian beam.