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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: D.F.Parker@ed.ac.uk
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Abstract

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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

References

[1]Achenbach, J. D. (1998) Explicit solutions for carrier waves supporting surface and plate waves. Wave Motion 28, 89–97.CrossRefGoogle Scholar
[2]Ash, E. A. & Paige, E. G. S. (1985) Rayleigh-Wave Theory and Application, Springer, New York.CrossRefGoogle Scholar
[3]Barnett, D. M. & Lothe, J. (1985) Free surface (Rayleigh) waves in anisotropic elastic half-spaces: The surface impedance method. Proc. R. Soc. Lond. A 402, 135–152.Google Scholar
[4]Biryukov, S. V., Gulyaev, Yu. V., Krylov, V. V. & Plessky, V. P. (1994) Surface Acoustic Waves in Inhomogeneous Media, Springer, New York.Google Scholar
[5]Chadwick, P. (1976) Surface and interfacial waves of arbitrary form in isotropic media. J. Elast. 6, 73–80.CrossRefGoogle Scholar
[6]Dai, H.-H., Kaplunov, J. & Prikazchikov, D. A. (2010) A long-wave model for the surface elastic wave in a coated half-space. Proc. Roy. Soc. Lond. A 466, 3097–3116.Google Scholar
[7]Dieulesaint, E. & Royer, D. (1986) Ondes Élastiques Dans les Solides, Masson, France.Google Scholar
[8]Gogoladze, V. G. (1948) Rayleigh waves on the interface between a compressible fluid medium and a solid elastic half-space. Trudy Seismolo. Inst. Acad. Nauk USSR 127, 27–32.Google Scholar
[9]Kaplunov, J., Nolde, E. & Prikazchikov, D. A. (2010) A revisit to the moving load problem using an asymptotic model for the Rayleigh wave. Wave Motion 47 (7), 440–451, doi:10.1016/j.wavemoti.2010.01.005CrossRefGoogle Scholar
[10]Kaplunov, J., Zakharov, A. & Prikazchikov, D. A. (2006) Explicit models for elastic and piezoelastic surface waves. IMA J. Appl. Math. 71, 768–782.CrossRefGoogle Scholar
[11]Kiselev, A. P. (2004) Rayleigh wave with a transverse structure. Proc. R. Soc. Lond. A 460, 3059–3064.CrossRefGoogle Scholar
[12]Kiselev, A. P. (2007) Localized light waves: Paraxial and exact solutions of the wave equation (a review). Opt. Spectrosc. 207, 661–681.Google Scholar
[13]Kiselev, A. P. & Parker, D. F. (2010) Omni-directional Rayleigh, Stoneley and Schölte waves with general time dependence. Proc. Roy. Soc. Lond. A 466, 2241–2258.Google Scholar
[14]Parker, D. F. (2009) Waves and statics for functionally graded materials and laminates. Int. J. Eng. Sci. 47, 1315–1321.CrossRefGoogle Scholar
[15]Parker, D. F. & Kiselev, A. P. (2009) Rayleigh waves having generalized lateral dependence. Quart. J. Mech. Appl. Math. 62, 19–29.CrossRefGoogle Scholar
[16]Parker, D. F. & Maugin, G. A. (1988) Recent Developments in Surface Acoustic Waves, Springer, New York.CrossRefGoogle Scholar
[17]Romeo, M. (2002) Uniqueness of the solution to the secular equation for viscoelastic surface waves. Appl. Math. Lett. 15, 649–653.CrossRefGoogle Scholar
[18]Rousseau, M. & Maugin, G. A. (2011) Rayleigh SAW and its canonically associated quasi-particle. Proc. Roy. Soc. Lond. A 467, 495–507.Google Scholar
[19]Schölte, J. G. (1947) The range of existence of Rayleigh and Stoneley waves. Mon. Not. R. Astron. Soc. Geophys. Suppl. 5, 120–126.CrossRefGoogle Scholar
[20]Stoneley, R. (1924) Elastic waves at the surface of separation of two solids. Proc. R. Soc. London A 106, 416–428.Google Scholar
[21]Strutt, J. W. (Lord Rayleigh) (1885) On waves propagated along the plane surface of an elastic solid. Proc. Lond. Math. Soc. 17, 4–11.Google Scholar
[22]Touhei, T. (2009) Generalized Fourier transform and its application to the volume integral equation for elastic wave propagation in a half-space. Int. J. Solids Struct. 46, 52–73.CrossRefGoogle Scholar