Green's Functions with Applications, 2nd Edition

Chapter 4

Green’s Functions for the Wave Equation

In Chapter 3, we showed how Green’s functions could be used to solve initial- and boundary-value problems involving ordinary differential equations. When we approach partial differential equations, similar considerations hold, although the complexity increases. In the next three chapters, we work through the classic groupings of the wave, heat and Helmholtz’s equations.

Of these three groups, we start with the wave equation

$\begin{array}{cc}{\nabla }^{2}u-\frac{1}{c^2}\frac{{\partial }^{2}u}{\partial t^2}=-q(\text{r},t),& (\mathbf{4.0.1})\end{array}$

where ∇ is the three-dimensional gradient operator, t denotes time, r is the position vector, c is the phase velocity of the wave, and q(r, t ...