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The Quarterly Journal of Mechanics and Applied Mathematics 2002 55(2):209-226; doi:10.1093/qjmam/55.2.209
© 2002 by Oxford University Press
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Application of Random Walk Methods to Wave Propagation

Bair V. Budaev1 and David B. Bogy1

( 1 University of California, Department of Mechanical Engineering, Berkeley, CA 94720, USA )

Here we present an approach to problems of diffraction that has its roots in a number of well-established theories such as the geometric theory of diffraction, the method of parabolic equations, the theory of Wiener functional integration, and the theory of stochastic processes. We start our analysis of the Helmholtz equation following closely the scheme of the ray method, but instead of approximating the resulting second-order auxiliary equation by a first-order equation, we study the original auxiliary equation and obtain its exact solution as the mathematical expectation of some functional in the space of Brownian trajectories with the Wiener probabilistic measure. The obtained solution appears to be a direct improvement over the ray method approximation to the exact solution of the Helmholtz equation, and it is shown to admit efficient numerical evaluation.

Received 24 July 2000. Revised 5 January and 4 March 2001.


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