TY - JOUR

T1 - Rayleigh waves having generalised lateral dependence

AU - Parker, D. F.

AU - Kiselev, A. P.

PY - 2009

Y1 - 2009

N2 - 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 eiθ, where θ ≡ k · x - ωt 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.

AB - 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 eiθ, where θ ≡ k · x - ωt 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.

UR - http://www.scopus.com/inward/record.url?scp=59549087030&partnerID=8YFLogxK

U2 - 10.1093/qjmam/hbn022

DO - 10.1093/qjmam/hbn022

M3 - Article

AN - SCOPUS:59549087030

VL - 62

SP - 19

EP - 30

JO - Quarterly Journal of Mechanics and Applied Mathematics

JF - Quarterly Journal of Mechanics and Applied Mathematics

SN - 0033-5614

IS - 1

ER -