1 The (one-dimensional) wave equation

The wave equation is a PDE which (as its name suggests) models wave-like phenomena. It is a model of waves on water, of sound waves, of waves of reactant in chemical reactions and so on. For the purposes of most of the following examples we may think of the application in hand as that of being a length of string tightly stretched between two points. Let $u=u\left(x,t\right)$ be the displacement from rest of the string at time $t$ and distance $x$ from one end. Oscillations in the string may be modelled by the wave equation

$u_{tt}=c^2{u}_{xx}\left(0<x<\mathrm{\&ell;},t>0\right)$

where $\mathrm{\&ell;}$ is the length of the string, $t=0$ is some initial time and $c>0$ is a constant (the wave speed ) dependent on the material properties of the string. (Further discussion of the constant $c$ is given in HELM booklet  25.2.)

The wave equation is hyperbolic, as we can readily verify on recalling the definitions at the beginning of Section 32.4. Extra information is needed to specify the initial value problem. The initial position and initial velocity are given as

$\left.\begin{array}{ccc}\hfill u\left(x,0\right)& \hfill =\hfill & f\left(x\right)\hfill \\ \hfill u_t\left(x,0\right)& \hfill =\hfill & g\left(x\right)\hfill \end{array}\right\}0\le x\le \mathrm{\&ell;}$

Finally, we need boundary conditions specifying how the ends of the string are held. For example

$u\left(0,t\right)=u\left(\mathrm{\&ell;},t\right)=0\left(t>0\right)$

models the situation where the string is fixed at each end.

(We will suppose that $f\left(0\right)=f\left(\mathrm{\&ell;}\right)=0$ so that there is no apparent conflict at the ends of the string at the initial time.)