Green's Functions with Applications, 2nd Edition

Chapter 6

Green’s Functions for the Helmholtz Equation

In the previous chapters, we sought solutions to the heat and wave equations via Green’s functions. In this chapter, we turn to the reduced wave equation

$\begin{array}{cc}{\nabla }^{2}u+\lambda u=-f\left(\mathbf{\text{r}}\right).& \left(\mathrm{6.0.1}\right)\end{array}$

Equation 6.0.1, generally known as Helmholtz’s equation, includes the special case of Poisson’s equation when λ = 0. Poisson’s equation has a special place in the theory of Green’s functions because George Green invented his technique for its solution.

The reduced wave equation arises during the solution of the harmonically forced wave equation1 by separation of variables. In one spatial dimension, the problem is