Diffraction by an impedance cone and Weyl-Van der Pol phenomenon in acoustics and electromagnetism

This paper deals with the asymptotic description of the diffraction pattern which is analogous 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. The incident wave illuminates a circular impedance cone. The singular point of the cone's boundary (the vertex of the cone) plays 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 the vertex along the impedance surface, may be located in a neighborhood of the saddle points. The latter are responsible for the spherical wave from the vertex of the cone. As a result, the asymptotics of the Sommerfeld integral are uniformly represented by a parabolic cylinder type function (the generalised Fresnel integral) for the cone problem.