The Weyl–Van Der Pol Phenomenon in Acoustic Diffraction by a Wedge or a Cone with Impedance Boundary Conditions

Abstract

The paper deals with the asymptotic description of a diffraction pattern similar to the classical Weyl–Van der Pol phenomenon (the Weyl–Van der Pol formula). The latter arises in the problem of diffraction of waves generated by a source located near an impedance plane. An incident wave illuminates an impedance wedge or cone. The singular points of the wedge’s (the edge points) or cone’s (the vertex of the cone) boundary play the role of an imaginary source, giving rise to a specific boundary layer in some neighborhood of the corresponding impedance surface, provided that the surface impedance is relatively small. From the mathematical point of view, the description of the phenomenon is given by means of the far field asymptotics for the Sommerfeld integral representations of the scattered field. For small impedance of the scattering surface, the singularities describing the surface wave, which propagates from the edge (or from the vertex) along the impedance surface, may be located in a neighborhood of saddle points. The latter are responsible for a cylindrical wave from the edge of the wedge (or for a spherical wave from the vertex of the cone). As a result, the asymptotics of the Sommerfeld integral are uniformly represented by a Fresnel type integral for the wedge problem or by a parabolic cylinder type function for the cone problem.