Weyl--Van der Poll phenomenon in acoustic diffraction by a~wedge or cone with impedance boundary conditions

M. A. Lyalinov

Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 45, Tome 438 (2015), pp. 178-202

Résumé

The work deals with the asymptotic description of the diffraction pattern which is analogous to the classical Weyl–Van der Poll phenomenon (the Weyl–Van der Poll formula). The latter arises in the problem of diffraction of waves generated by a source located near an impedance plane. The 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 the specific boundary layer in some vicinity of the corresponding impedance surface provided 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 the 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 the saddle points. The latter are responsible for the cylindrical wave from the edge of the wedge (or for the 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.