On the shielding of sound from a source near one or two coaxial cones

Abstract

The shielding of the sound from a source near a cone or two coaxial circular cones is addressed by an exact analytical method, considering free spherical waves satisfying a rigid or impedance boundary condition on the wall of the cone(s); the boundary condition is applied for impedance proportional to the distance from the vertex. The exact solution involves the eigenvalues of the problem that are generally real or complex (not integers), and coincide with the degree of the associated Legendre functions and order of the spherical Bessel functions which specify respectively the latitudinal and radial dependencies of the wave field. These eigenfunctions or ‘conical wave harmonics’ can be chosen in more than one way, e.g. (a) as standing wave modes which are finite at the vertex of the cone(s), but do not satisfy a radiation condition at infinity; (b) as propagating waves satisfying a radiation condition at infinity, and generally singular at the vertex of the cone(s). The method of calculation of eigenvalues and eigenfunctions is presented both for a single cone and two coaxial cones with the same vertex, and arbitrary aperture(s). The method specifies the eigenvalues, and the corresponding radial, azimuthal and latitudinal eigenfunctions. An asymptotic formula is obtained for the eigenvalues which gives reasonable good agreement with the exact results. The eigenvalues and eigenfunctions appear with suitable amplitudes in the Green function representing a monopole source near the vertex of the cone(s). The acoustic field is plotted also for a longitudinal and a transverse dipole and mixed quadrupole source near the vertex.