Diffraction of a Plane Acoustic Wave from a Finite Soft (Rigid) Cone in Axial Irradiation

Abstract

The problem of diffraction of a plane acoustic wave by a finite soft (rigid) cone is investigated. This one is formulated as a mixed boundary value problem for the three-dimensional Helmholtz equation with Dirichlet (Neumann) boundary condition on the cone surface. The diffracted field is sought as expansion of unknown velocity potential in series of eigenfunctions for each region of the existence of sound pressure. The solution of the problem then is reduced to the infinite set of linear algebraic equations (ISLAE) of the first kind by means of mode matching technique and orthogonality properties of the Legendre functions. The main part of asymptotic of ISLAE matrix element determined for large indexes identifies the convolution type operator amenable to explicit inversion. This analytical treatment allows one to transform the initial diffraction problem into the ISLAE of the second kind that can be readily solved by the reduction method with desired accuracy depending on a number of truncation. All these determine the analytical regularization method for solution of wave diffraction problems for conical scatterers. The boundary transition to soft (rigid) disc is considered. The directivity factors, scattering cross sections, and far-field diffraction patterns are investigated in both soft and rigid cases whereas the main attention in the near-field is focused on the rigid case. The numerically obtained results are compared with those known for the disc.