Eigenfunction expansions of solutions to the Helmholtz equation

The “solutions” of boundary-value problems presented in Section 2.8 are merely formal in that the appropriate Green functions must be found, and finding these Green functions is extremely difficult except in certain special cases such as the Rayleigh-Sommerfeld problem. Moreover, even when the appropriate Green functions can be found they are often expressed in the form of superpositions of elementary solutions of the homogeneous Helmholtz equation that are especially suited for dealing with boundaries of a specific shape. One example of a set of such elementary solutions is the plane waves which arise from applying the method of separation of variables to the homogeneous Helmholtz equation using a Cartesian coordinate system and, as we will see below, form a complete set of basis functions for fitting boundary-value data specified on plane surfaces; e.g., for the RS problems. However, the plane waves have limited utility in solving boundary-value problems involving non-planar boundaries such as spherical boundaries. In that case the method of separation of variables is applied to the Helmholtz equation using a spherical polar coordinate system and the so-called multipole fields arise as a set of elementary solutions that form a basis for fitting boundary-value data specified on spherical boundaries. In this chapter we will briefly review the method of separation of variables for the Helmholtz equation and obtain the resulting eigenfunctions for the important cases of Cartesian, spherical polar and cylindrical coordinate systems.