## 1. Introduction

[2] For more than a century problems of wave diffraction by an infinite wedge have been at the center of the mathematical theory of diffraction starting from the landmark papers of *Poincaré* [1892, 1897] and *Sommerfeld* [1896], who presented explicit descriptions of wavefields generated by a plane incident wave in an infinite wedge with ideally reflecting faces. However, they used very different methods. The solution of Poincaré was based on the method of separation of variables, and it led to a solution represented by an infinite series of Bessel functions multiplied by associated trigonometric polynomials. The solution of Sommerfeld was in terms of the so-called Sommerfeld integral representing wavefields as superpositions of plane waves. Although these two solutions look very different, they are equivalent and can be easily converted from one to the other. It took about fifty years before a problem of diffraction by a wedge with impedance boundary conditions was solved. In the 1950s this problem was independently solved by *Maliuzhinets* [1951, 1955], *Senior* [1959] and *Williams* [1959]. They used techniques that are similar in general but different in detail that employed Sommerfeld's representation of wavefields to reduce the problem to certain functional equations in the complex plane. In the following decades the Maliuzhinets method was applied to a number of two- and three-dimensional problems of diffraction by infinite wedges. However, all of these problems either had boundary conditions with constant coefficients or even simpler boundary conditions with the coefficients proportional to the distance from the vertex, as considered by *Felsen and Marcuvitz* [1972].

[3] Here we consider a two-dimensional problem of diffraction by a wedge with a variable impedance at its face. We employ a novel probabilistic approach to wave propagation that is not restricted to problems with simple canonical geometries or to problems with special (constant) boundary conditions. The probabilistic approach has already been successfully applied by the authors to a standard two-dimensional problem of diffraction by a wedge with constant impedances, as well as to more challenging three-dimensional problems of diffraction by a plane angular sector [*Budaev and Bogy*, 2004] and by an infinite wedge with anisotropic face impedances [*Budaev and Bogy*, 2006b]. Most recently, the probabilistic method was applied to diffraction by an arbitrary convex polygon with side-wise constant impedances [*Budaev and Bogy*, 2006a]. This problem with nontrivial geometry does not have known conventional closed-form solutions, but its probabilistic solution is not considerably more complex than the solutions of the other problems mentioned above. Here we take the next step and extend the probabilistic method in a way which makes it possible to handle boundary conditions with variable coefficients. This is the centerpiece of the paper, and it is based on further development of the idea of analytical continuation introduced by *Budaev and Bogy* [2005c] and *Budaev and Bogy* [2006a], who applied it to other problems of interest.

[4] The paper is organized as follows. In section 2 we formulate the problem and convert it to a form suitable for the application of the random walk method. Section 3 deals with the problem with constant coefficients. Most of this material is published by *Budaev and Bogy* [2005b], but it is included here also to make the presentation of section 4 transparent and self-consistent. The main material of the paper is concentrated in section 4, and section 5 presents numerical experiments based on the obtained solution.