Study of field source structures described by zonal spherical wave functions

Abstract

The structures of scalar field sources described by zonal spherical wave functions have been studied. Such functions are used, in particular, in problems of electrodynamics and acoustics when a spherical surface limits the domain. The values of the fields under consideration depend not only on the distance to the source, but also on the angular coordinate, which indicates the multipole nature of such a source. Therefore, the study of their structures is an interesting theoretical problem.

A multipole source is a set of monopoles (zero-order field sources) located within a specific infinitely small region. In this study, an expression has been first obtained that relates the zero-order field to the fields of subsequent orders. The differential nature of this expression shows that the size of the multipole, unlike the dimensionless monopole, is an infinitesimally small quantity. By assigning this quantity small finite values, it is demonstrated that monopoles in a multipole are located along a short segment. The amplitudes of these monopoles are also found. The resulting multipole is called linear. Next, based on the well-known addition theorem, other values of monopole amplitudes are determined, at which the initial field corresponding to the zonal spherical wave function is described more accurately. A generalization of the linear multipole for the case of a larger number of monopoles is also given. Based on the same addition theorem, it is further shown that the initial field can also be described by monopoles, which, unlike the previous cases, are located on the surface of a small radius sphere.