Introduction

The wave equation, which in one dimension takes the form

$$\frac{\partial^2\xi}{\partial t^2} = c^2\,\frac{\partial^2 \xi}{\partial x^2},$$ (230)

occurs so frequently in physics that it is not necessary to enumerate examples. Here, $\xi$ is usually some sort of displacement or perturbation, whereas $c$ is the (constant) wave speed. The wave equation possesses the formal solution

$$\xi(x,t) = F(x-c\,t) + G(x+c\,t),$$ (231)

where $F$ and $G$ are arbitrary functions. The above solution represents arbitrarily shaped wave pulses propagating with speed $c$ in the $+x$ and $-x$ directions, respectively, without changing shape.

The wave equation, which is second-order in space and time, can be written as two coupled first-order equations by defining the new variables $v=\partial\xi/\partial t$ and $\theta=-c\,\partial\xi/\partial x$. Expressing Eq. (230) in terms of these new variables, we obtain

$$\frac{\partial v}{\partial t} + c\,\frac{\partial \theta}{\partial x} = 0,$$ (232)

$$\frac{\partial \theta}{\partial t} + c\,\frac{\partial v}{\partial x} = 0.$$ (233)

Note that when solving the wave equation numerically it is generally preferable to write it as a set of coupled first-order equations, as shown above.