Neumann Problem Three - Dimensional Helmholtz Equation

A method for explicitly solving the exterior Dirichlet problem for the threedimensional Helmholtz equation in terms of the Dirichlet Green's function for Laplace's equation has recently been found [6]. The present work shows how a similar technique may be used to solve the exterior Neumann problem in terms of the corresponding Neumann-Green function for Laplace's equation. The existence of the required potential Green's function is proven by KELLOGG [5]. This work formed a basis for WEVL [14] and MiJLLER [9] who proved that a solution of the exterior Dirichlet problem for the Helmholtz equation exists. LEIS [7] extended these ideas to establish the existence of a solution of the exterior Neumann problem for the Helmholtz equation and an alternative existence proof has recently been given by WER~R [13]. The utility of these results in actually producing a solution, however, has not been demonstrated. That the solution of the boundary value problem for the Helmholtz equation may be found as a perturbation of the solution of the corresponding potential problem is shown by NOBLE [10]. The solution is expressed as a power series in wave number, each term of which is the solution of an integral equation of the second kind which differs from term to term only in its inhomogeneous part. This formulation of the problem does not yield an explicit representation of the solution, however, except as the formal inverse of an integral operator. In the present work, a representation of the desired solution is derived which expresses the solution as a linear operation on itself plus a known term. The linear operator is shown to be bounded, and the inverse, for small values of wave number, is given by a standard LiouviUe-Neumann series. The principal result of the paper is contained in the following.