The decay of solutions of the two dimensional wave equation in the exterior of a straight strip

Wolfe, Peter

doi:10.1090/S0002-9947-1977-0442488-8

The decay of solutions of the two dimensional wave equation in the exterior of a straight strip
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Trans. Amer. Math. Soc. 231 (1977), 405-428 Request permission

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 $O(1/t)$ as $t \to \infty$. 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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