Wave Motion 40 (2004) 163–172 Wave scattering by surface-breaking cracks and cavities Bair V. Budaev ∗ , David B. Bogy Department of Mechanical Engineering, University of California, Berkeley, CA 94720-1740, USA Received 18 July 2003; received in revised form 15 January 2004; accepted 6 February 2004 Available online 16 March 2004 Abstract Here we suggest a new approach to the analysis of two-dimensional wave scattering by arbitrarily shaped irregularities of a boundary of the half-space. This approach employs the probabilistic method of a rigorously justified representation of the scattered waves by explicit formulas involving the computation of mathematical expectations which average the values of certain functionals computed along the trajectories of the random motion governed by specified stochastic differential equations. As typical for the random walk method, the obtained solutions admit numerical implementations by simple algorithms which are practically independent of the particular geometry of the scatterer, which are inexpensive in terms of computer memory requirements, and which have virtually unrestricted capability for parallel processing. Illustrative numerical examples include problems of wave scattering by a tilted straight crack and by a circular cavity. © 2004 Elsevier B.V. All rights reserved. 1. Introduction The diffraction of waves by a localized irregularity of a flat boundary of the half-space is a phenomenon of considerable interest for various branches of science and engineering including, but not limited to, electromagnetic theory, acoustics, and non-destructive evaluation. However, despite its considerable attention, the arsenal of methods available for the quantitative analysis of this phenomenon remains relatively limited even in the cases that may be formulated as two-dimensional scalar problems for the Helmholtz equations with a constant wave number. Most of the results available in this area are related to the plane problem of wave scattering by a straight crack orthogonal to the boundary of the half-plane, which is explained by the possibility of symmetrical continuation of such a configuration to the entire plane with a straight slit. The simple geometry of the slit makes it possible to describe waves generated in its exterior by a number of distinctively different methods. Thus, the observations that the Helmholtz equation admits separation of variables in elliptic coordinates and that the slit is an infinitely thin ellipse suggest that the scattered wave can be represented by an infinite series with the terms expressed through Mathieu functions. Another approach explores the physically clear idea of multiple diffractions by the slit’s edges. This idea has been implemented by various techniques resulting in exact or approximate solutions tailored for specific needs, such as for the analysis of scattering by a short or, oppositely, by a long crack. To avoid bias in citing original work dispersed in a vast literature covering very different fields, we mention only the collection [1, Chapter 4] which presents a bibliography and review of many important contributions to the topic. ∗ Corresponding author. Tel.: +1-510-6435776; fax: +1-510-6436162. E-mail addresses: budaev@me.berkeley.edu, budaev@cml.me.berkeley.edu (B.V. Budaev), dbogy@cml.me.berkeley.edu (D.B. Bogy). 0165-2125/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.wavemoti.2004.02.003