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Evanescent Schölte waves of arbitrary profile and direction

Published online by Cambridge University Press:  09 November 2011

D. F. PARKER
D. F. PARKER*
Affiliation:
School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, Edinburgh, EH9 3JZ, U.K. email: [email protected]
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Abstract

Schölte waves, waves bound to the interface between a fluid and an elastic half-space, are, for many material combinations, evanescent; as they propagate, they are damped due to radiation. A representation of the general evanescent Schölte wave is here obtained in terms of a solution to the membrane equation with complex speed, linked, at each instant, to a complex-valued harmonic function in a half-space. This derivation generalises one obtained recently for (non-evanescent) Rayleigh, Stoneley and Schölte waves. An alternative description is also obtained, in which the time-evolution of the normal displacement of the interface satisfies a first-order, complex-valued, non-local evolution equation. Amongst some explicit solutions obtained are decaying solutions allied to a general solution to the Helmholtz equation, and a solution closely related to a Gaussian beam. In the plane–strain case, the general Schölte wave splits into two disturbances, one right-travelling and one left-travelling, each being described at all times in terms of a harmonic function in a half-plane, decaying with depth yet having arbitrary boundary values. This representation highlights the dual elliptic–hyperbolic nature typical of guided waves and gives a surprisingly compact representation for the two-dimensional case.

Keywords

Schölte wavesevanescentmembrane equationcomplex speed
Type
Papers
Information
European Journal of Applied Mathematics , Volume 23 , Issue 2 , April 2012 , pp. 267 - 287
Copyright
Copyright © Cambridge University Press 2011

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