Diffraction by a convex polygon with side-wise constant impedance Bair V. Budaev * , David B. Bogy Department of Mechanical Engineering, University of California, Berkeley, CA 94720, USA Received 13 January 2006; received in revised form 9 May 2006; accepted 25 May 2006 Available online 1 August 2006 Abstract The recently developed probabilistic approach to wave propagation and diffraction is applied to the two-dimensional problem of diffraction by an arbitrary convex polygon with side-wise constant impedance boundary conditions. Surpris- ingly, the new method provides a rigorous solution which is simple, transparent and complimentary to the ray approxi- mation. It is easy to implement and it is compatible with intuitive ideas about diffraction. The numerical examples confirm the feasibility of the solution. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Wave propagation; Diffraction; Ray method; Brownian motion; Stochastic processes 1. Introduction The theory of wave propagation has traditionally considered problems of diffraction by simplified objects which permit exact closed-form solutions. The first problem of this kind was solved in the 1890s, when Poincare ´ [11,12] and Sommerfeld [13] independently explicitly described the waves generated by a plane inci- dent wave in an infinite wedge with ideally reflecting faces. After that, many other problems of diffraction were solved analytically, and the interest in such solutions was boosted by the development of the geometrical theory of diffraction [9], which heavily uses the diffraction coefficients derived from the analysis of wave fields in canonical domains. The pioneering papers of Poincare ´ and Sommerfeld not only presented explicit solutions of the diffraction problem, but they also presented two different approaches. Thus, Poincare ´ employed the method of separation of variables and Sommerfeld represented the solution by the so-called Sommerfeld integral, which was equiva- lent to using the integral Mellin transform. For more than a century the methods of separation of variables and integral transforms have remained the primary approaches to closed-form solutions in the theory of diffraction, which remarkably confirms the power of these methods, but which also explains why closed-form solutions are 0165-2125/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.wavemoti.2006.05.007 * Corresponding author. Tel.: +1 510 643 5776; fax: +1 510 642 6163. E-mail addresses: budaev@berkeley.edu (B.V. Budaev), dbogy@cml.me.berkeley.edu (D.B. Bogy). Wave Motion 43 (2006) 631–645 www.elsevier.com/locate/wavemoti