Numerical Calculation of the Solution of the Helmholtz Equation on the Sphere

Abstract:
The well-known solution of the Helmholtz equation for a point source above a sphere with Dirichlet boundary condition is considered. This solution consists of a series containing Legendre functions and spherical Bessel functions. A method for a numerical approximation to this series is described. The method also applies to spheres with large radii, as for example the spherical earth. The computation time on a PC for it is in an acceptable range for broadcast frequencies. The method applies both to the line-of-sight and to the shadow region of the sphere. The results for the shadow region are confirmed by comparing them with results of the Watson transformation of the solution. Results for the line-of-sight region show that the concept of a divergence factor D, where D originated from the geometrical optics limit for the sphere, is not applicable to frequencies as low as the ones in terrestrial broadcast networks, and to the usual situation of grazing incidence in these networks. The result of the paper, the method for a numerical approximation to the series, can be used to check propagation models for mobile receivers, as e.g. automotive broadcast receivers. It can also be used in various areas of physics and engineering, particularly in acoustics, to check or to provide numerical results of solutions of the Helmholtz equation on the sphere.