Scattering of acoustic waves by a sector

In this work we study the problem of diffraction of an acoustic plane wave by a plane angular sector with the Dirichlet boundary condition on its surface. By means of incomplete separation of variables, with the aid of Watson–Bessel integral representation the problem is reduced to an infinite system of linear summation equations of the second kind. Exploiting the reduction of the integral representation to that of the Sommerfeld type, a consequent procedure is then developed in order to describe different components in the far field asymptotics. To that end, the analytic properties and singularities of the integrand in the Sommerfeld integral are carefully studied. The latter plays a crucial role when evaluating the Sommerfeld integral by means of the saddle point technique because these singularities are captured in the process of deformation of the Sommerfeld contours into the steepest descent paths. The corresponding asymptotic contributions of the singularities lead to the description of the different types of waves in the far field asymptotics. These are the waves reflected from the sector, the waves from the edges including those multiply diffracted from one edge to another. The spherical wave from the vertex of the sector is specified by the saddle points. The singularities migrate, provided the observation point moves, and may coalesce with each other and with the saddle points, which requires more accurate asymptotic evaluation of the Sommerfeld integral in terms of the transition special functions closely related to the Fresnel integral.