The Incidence of the Plane Wave on an Elastic Wedge at a Critical Angle

The problem of diffraction by an elastic wedge has a significant history. A rather detailed survey was presented in [1], so we shall restrict ourselves to a short commentary. Up to the mid 1990s, most of the authors had been interested in the computational aspects of the problem, leaving the question of rigorous formulation aside (we mean the existence and uniqueness theorems). In the mid 1990s, a seminal approach, called the spectral function method, was worked out by G. Lebeau; this approach makes it possible to treat the diffraction problems in “wedge-like domains” and does not involve separation of variables. In particular, in [2] this method was applied to the problem of an elastic wedge immersed in a fluid: the solvability of the plane wave diffraction problem was proved, and its solution was represented in the form of a single layer potential. In [3], the same approach was applied to the elastic wedge problem. The existence and uniqueness of the solution was proved in the class of functions satisfying the radiation conditions formulated in [3]. The case of the critical incidence, i.e., the situation where a grazing plane wave traveling to the wedge tip appears, was not considered in [2] and [3]. Our aim in the present paper is to fill this gap; we prove the existence and uniqueness for the critical case as well.