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The propagation of Rayleigh waves over curved surfaces at high frequency

Published online by Cambridge University Press:  24 October 2008

R. D. Gregory
R. D. Gregory
Affiliation:
Department of Mathematics, University of Manchester
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Abstract

A formal asymptotic theory, valid at high frequencies, is developed for the propagation of time harmonic Rayleigh surface waves over the general smooth free surface Σ of a homogeneous elastic solid. It is shown that on Σ these Rayleigh waves can be described by a system of surface rays, which are shown to be geodesics of Σ. The amplitude of the waves on Σ is shown to vary in such a way that the energy propagated along a strip of surface rays is constant. The waves are also shown to be dispersive and an explicit first-order dispersion formula is derived.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 70 , Issue 1 , July 1971 , pp. 103 - 121
Copyright
Copyright © Cambridge Philosophical Society 1971

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References

REFERENCES

(1)Babich, V. M. and Rusakova, N. YA.The propagation of Rayleigh waves over the surface of a non-homogeneous elastic body with an arbitrary form. U.S.S.R. Comput. Math. and Math. Phys. 2 (1962), 652–665.Google Scholar
(2)Biot, M. A.Propagation of elastic waves in a cylindrical bore containing a fluid. J. Appl. Phys. 23 (1952), 997–1005.Google Scholar
(3)Bolt, B. A. and Dorman, J.Phase and group velocities of Rayleigh waves in a spherical gravitating earth. J. Geophys. Res. 66 (1961), 2965–2981.CrossRefGoogle Scholar
(4)Erdélyi, A. et al. Tables of integral transforms, Vol. I (McGraw-Hill, 1954).Google Scholar
(5)Ewing, W. M., Jardetzky, W. S. and Press, F.Elastic waves in layered media (McGraw-Hill, 1957).CrossRefGoogle Scholar
(6)Gregory, R. D.The attenuation of a Rayleigh wave in a half-space by a surface impedance. Proc. Cambridge Philos. Soc. 62 (1966), 811–827.Google Scholar
(7)Grimshaw, R.Propagation of surface waves at high frequencies. J. Inst. Math. Appl. 4 (1968), 174–193.Google Scholar
(8)Hayes, M. and Rivlin, R. S.A note on the secular equation for Rayleigh waves. Z. Angew. Math. Phys. 13 (1962), 80–83.CrossRefGoogle Scholar
(9)Karal, F. C. and Keller, J. B.Elastic wave propagation in homogeneous and inhomogeneous media. J. Acoust. Soc. Amer. 31 (1959), 694–705.Google Scholar
(10)Keller, J. B. and Karal, F. C.Geometrical theory of elastic surface-wave excitation and propagation. J. Acoust. Soc. Amer. 36 (1964), 32–40.CrossRefGoogle Scholar
(11)Keller, J. B., Lewis, R. M. and Seckler, B. D.Asymptotic solution of some diffraction problems. Comm. Pure Appl. Math. 9 (1956), 207–265.Google Scholar
(12)Rulf, B.Rayleigh waves on curved surfaces. J. Acoust. Soc. Amer. 45 (1969), 493–499.CrossRefGoogle Scholar
(13)Smith, R. Asymptotic solutions for high frequency trapped wave propagation. University of Bristol Ph.D. Thesis (1970).Google Scholar
(14)Titchmarsh, E. C.Introduction to the theory of Fourier integrals (Oxford Clarendon Press, 1937).Google Scholar
(15)Viktorov, I. A.Rayleigh-type waves on a cylindrical surface. Soviet Physics Acoust. 4 (1958), 131–136.Google Scholar
(16)Viktorov, I. A.Rayleigh and Lamb waves (Plenum Press, New York, 1967).Google Scholar
(17)Watson, G. N.A treatise on the theory of Bessel functions, 2nd ed. (Cambridge, 1944).Google Scholar