Definition

The one-dimensional wave equation is defined as

$$u_{tt} = c^{2}u_{xx}$$ (6.1)

It is a second-order partial differential equation in the dependent variable $u=u(x,t)$, where $x$ is a variable representing distance, and $t$, as before is time. The equation is defined over $t\geq 0$, and for $x\in{\mathcal D}$, where ${\mathcal D}$ is some simply connected subset of ${\mathbb{R}}$. $c$ is the wave speed.

As mentioned above, the wave equation is a simple first approximation, under low amplitude conditions, to the transverse motion of strings (in which case $u$ is the transverse displacement, and $c=\sqrt{T/\rho}$, where $T$ is the applied string tension and $\rho$ is the linear mass density), to the longitudinal motion of bars, and to longitudinal vibration of an air column in a tube of uniform cross-section. Under musical conditions, it is rather a better approximation in the latter cases than in the former; string nonlinearities which lead to perceptually important effects will be dealt with in detail in Chapter 8.

Figure: Different physical systems which are solved by the wave equation (6.1): (a), a lossless string vibrating at low-amplitude, where $u(x,t)$ represents the string displacement; (b) an acoustic tube, under lossless conditions, and for which $u(x,t)$ represents the deviation in pressure about a mean; (c) a lossless transmission line, where $u(x,t)$ represents the voltage between the pair of lines.