The decay of solutions of the two dimensional wave equation in the exterior of a straight strip
Author:
Peter Wolfe
Journal:
Trans. Amer. Math. Soc. 231 (1977), 405428
MSC:
Primary 35L05
MathSciNet review:
0442488
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Abstract: We study an initial boundary value problem for the wave equation in the exterior of a straight strip. We assume the initial data has compact support and that the solution vanishes on the strip. We then show that at any point in space the solution is as . This is the same rate of decay as obtains for the solution of the initial boundary value problem posed in the exterior of a smooth star shaped region. Our method is to use a Laplace transform. This reduces the problem to a consideration of a boundary value problem for the Helmholtz equation. We derive estimates for the solution of the Helmholtz equation for both high and low frequencies which enable us to obtain our results by estimating the Laplace inversion integral asymptotically.
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 P. D. Lax, C. S. Morawetz and R. S. Phillips, Exponential decay of solutions of the wave equation in the exterior of a starshaped obstacle, Comm. Pure Appl. Math. 16 (1963), 477486. MR 27 #5033. MR 0155091 (27:5033)
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 C. S. Morawetz, The limiting amplitude principle, Comm. Pure Appl. Math. 15 (1962), 349361. MR 27 #1696. MR 0151712 (27:1696)
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 L. A. Muraveĭ, The decrease of solutions of the second exterior boundaryvalue problem for the wave equation with two space variables, Dokl. Akad. Nauk SSSR 193 (1970), 996999 = Soviet Math. Dokl. 11 (1970), 10671071. MR 42 #3400. MR 0268503 (42:3400)
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 G. Okikiolu, Aspects of the theory of bounded integral operators in spaces, Academic Press, New York, 1971. MR 0445237 (56:3581)
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 V. G. Sologub, The solution of a certain integral equation of convolution type with finite limits of integration, Ž. Vyčisl. Mat. i Mat. Fiz. 11 (1971), 837854 = USSR Comput. Math. and Math. Phys. 11 (1971), no. 4, 3352. MR 45 #832. MR 0291741 (45:832)
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 , Diffraction of plane waves by a strip; exact and asymptotic solutions, SIAM J. Appl. Math. 23 (1972), 118132. MR 47 #626. MR 0312064 (47:626)
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 , Low frequency diffraction by a hard strip, SIAM J. Appl. Math. 29 (1975), 273287. MR 0387804 (52:8643)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197704424888
PII:
S 00029947(1977)04424888
Keywords:
Wave equation,
rate of decay of solutions,
asymptotic behavior of solutions,
Laplace transform,
reduced wave equation
Article copyright:
© Copyright 1977
American Mathematical Society
