Wave Equation in Higher Dimensions

The wave equation in 1D, 2D, or 3D may be written as

$$\left(\nabla ^2 - \frac{1}{c^2}\frac{\partial^2}{\partial t^2} \right) z(\underline{x},t) \eqsp 0, \protect$$ (B.49)

where, in 3D, $z(\underline{x},t)$ denotes the amplitude of the wave at time $t$ and position $\underline{x}\in\mathbb{R}^3$ , and

$$\nabla ^2 \isdefs \nabla \cdot \nabla \isdefs \nabla ^T\nabla \eqsp \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}$$

denotes the Laplacian operator in Euclidean coordinates. (In general coordinates, it is often denoted by $\Delta$ .) To investigate solutions of the wave equation, as pursued in §B.8.3 below, it is useful to first develop some simple expressions and notations for elementary waves in 2D and 3D.