Spherical Sound Sources and Line Arrays

Abstract

The wave equation, $\nabla^2 p = \ddot{p} / c^2$ as derived in Sections 7.1 and 7.2 theoretically determines all possible sound fields in idealized fluids, that is, gases and liquids. The special task of computing sound fields for particular cases requires solutions of the wave equation for particular boundary conditions. In general, this task can be mathematically expensive, but there are helpful computer programs available, some of which are based on numerical methods like the finite-element method, FEM, or the boundary-element method, BEM. In praxi, approximations are often sufficient to understand the structure of a problem.

Closed solutions of the wave equation only exist for a limited number of special cases. We have already introduced the plane wave as one-dimensional solution in Cartesian coordinates. A few further one-, two- and three-dimensional cases are solvable in closed form, especially when symmetries allow for simplified formulations using appropriate coordinate systems as is the case for spherical or cylindrical coordinates.