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Elastic waves in two solids as propagation of singularities phenomenon

Published online by Cambridge University Press:  22 January 2016

Kazuhiro Yamamoto
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In this paper we shall study elastic waves in two isotropic media with different densities and Lamè’s constants. In seismology elastic waves are studied when the border of two media is a hyperplane, however there are no results on elastic waves in the non-flat border case. First in Section 1 we shall show an existence theorem of the solutions of an initial boundary value problem which is satisfied by the displacements of two media. Next we shall discuss about propagation of singularities of the solutions, for Hörmander and Lax-Nirenberg showed that an appearance of propagation of singularities is similar to one of propagation of waves.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 116 , December 1989 , pp. 25 - 42
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1989

References

[1] Duvaut, G. and Lions, J. L., Inequalities in mechanics and physics, Grundlehren d. Math. Wiss., 219, Springer-Verlag, 1976.Google Scholar
[2] Hörmander, L., Linear partial differential operators, Grundlehren d. Math. Wiss., 116, Springer-Verlag, 1963.Google Scholar
[3] Hörmander, L., Fourier integral operator I, Acta Math., 127 (1971), 79–183.Google Scholar
[4] Hörmander, L., Fourier integral operators II, Acta Math., 128 (1972), 183–269.Google Scholar
[5] Nirenberg, L., Lectures on linear partial differential equations, Regional Conferences Series in Math., 17, Providence 4, R.I., 1973.Google Scholar
[6] Melrose, R. and Sjostrand, J., Singularities of boundary value problems I, Comm. Pure Appl. Math., 31 (1978), 593–617.CrossRefGoogle Scholar
[7] Stoneley, R., Elastic waves at the surface of separation of two solids, Proc. Roy. Soc. London, 106 (1924), 416–428.Google Scholar
[8] Taylor, M. E., Reflection of singularities of solutions to systems of differential equations, Comm. Pure Appl. Math., 28 (1975), 457–478.CrossRefGoogle Scholar
[9] Yamamoto, K., Singularities of solutions to the boundary value problems for elastic and Maxwell’s equations, Japanese J. Math., 14 (1988), 119–163.Google Scholar
[10] Yamamoto, K., Reflective elastic waves at the boundary as a propagation of singularities phenomenon, (to appear).Google Scholar
[11] Wilcox, C. H., Scattering Theory for the d’Alembert Equation in Exterior Domains, Lecture Notes in Math., 442, Springer-Verlag, 1975.Google Scholar