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March 2010 Singular Support of the Scattering Kernel for the Rayleigh Wave in Perturbed Half-Spaces
Mishio Kawashita, Hideo Soga
Methods Appl. Anal. 17(1): 1-48 (March 2010).


This paper deals with the Rayleigh wave scattering on perturbed half-spaces in the framework of the Lax-Phillips type. Singular parts of the scattering kernel for this scattering are closely connected with singularities of the Rayleigh wave passing through the perturbation on the boundary. This can be described by estimating the singular support of the scattering kernel on the Rayleigh wave channel. The proof is based on a representation formula of the scattering kernel that was obtained in the previous work. However, the formula does not suit the situation of the Rayleigh wave, even though it is a natural extension of Majda’s formula for the usual wave equation. Hence, the formula needs to be reformed, and the problem needs to be reduced to a pseudo-differential equation on the boundary governing the Rayleigh wave. Key methods for the reduced problem are construction of an approximate solution for the Rayleigh wave and analysis of an oscillatory integral distilled by using the solution. The phase function of the oscillatory integral is always degenerate along the characteristic curve of the Rayleigh wave. This degeneracy is handled by introducing a certain criterion for the regularity of the distribution defined by the oscillatory integral.


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Mishio Kawashita. Hideo Soga. "Singular Support of the Scattering Kernel for the Rayleigh Wave in Perturbed Half-Spaces." Methods Appl. Anal. 17 (1) 1 - 48, March 2010.


Published: March 2010
First available in Project Euclid: 6 December 2010

zbMATH: 1209.35075
MathSciNet: MR2735099

Primary: 35L20 , 35P25 , 74B05

Keywords: distorted plane waves , elastic wave equations , scattering kernel , scattering theory , the Rayleigh wave

Rights: Copyright © 2010 International Press of Boston

Vol.17 • No. 1 • March 2010
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