1 Wave equation

The simplest situation to give rise to the one-dimensional wave equation is the motion of a stretched string - specifically the transverse vibrations of a string such as the string of a musical instrument. Assume that a string is placed along the $x$ -axis, is stretched and then fixed at ends $x=0$ and $x=L$ ; it is then deflected and at some instant, which we call $t=0$ , is released and allowed to vibrate. The quantity of interest is the deflection $u$ of the string at any point $x$ , $0\le x\le L$ , and at any time $t>0$ . We write $u=u\left(x,t\right)$ . Figure 2 shows a possible displacement of the string at a fixed time $t$ .

Figure 2

Subject to various assumptions $\cdots$

1. damping forces such as air resistance are negligible
2. the weight of the string is negligible
3. the tension in the string is tangential to the curve of the string at any point
4. the string performs small transverse oscillations i.e. every particle of the string moves strictly vertically and such that its deflection and slope as every point on the string is small.

$\cdots$ it can be shown, by applying Newton’s law of motion to a small segment of the string, that $u$ satisfies the PDE

$\frac{{\partial }^{2}u}{\partial t^2}=c^2\frac{{\partial }^{2}u}{\partial x^2}$ (1)

where $c^2=\frac{T}{\rho }$ , $\rho$ being the mass per unit length of the string and $T$ being the (constant) horizontal component of the tension in the string. To determine $u\left(x,t\right)$ uniquely, we must also know

1. the initial definition of the string at the time $t=0$ at which it is released
2. the initial velocity of the string.

Thus we must be given initial conditions

$u\left(x,0\right)=f\left(x\right)0\le x\le L$ (initial position)

$\frac{\partial u}{\partial t}\left(x,0\right)=g\left(x\right)0\le x\le L$ (initial velocity)

where $f\left(x\right)$ and $g\left(x\right)$ are known.These two initial conditions are in addition to the two boundary conditions

$u\left(0,t\right)=u\left(L,t\right)=0$ for $t\ge 0$

which indicate that the string is fixed at each end. In Example 2 discussed in Section 25.1 we had

$f\left(x\right)=u_{0}sin\left(\frac{\pi x}{\ell }\right)$

$g\left(x\right)=0$ (string initially at rest).

The PDE (1) is the (undamped) wave equation. We will discuss solutions of it for various initial conditions later. More complicated forms of the wave equation would arise if some of the assumptions were modified. For example:

1. $\frac{{\partial }^{2}u}{\partial t^2}=c^2\frac{{\partial }^{2}u}{\partial x^2}-g$   if the weight of the string was allowed for,
2. $\frac{{\partial }^{2}u}{\partial t^2}=c^2\frac{{\partial }^{2}u}{\partial x^2}-\alpha \frac{\partial u}{\partial t}$ if a damping force proportional to the velocity of the string

   (with damping constant $\alpha$ ) was included.

Equation (1) is referred to as the one-dimensional wave equation because only one space variable, $x$ , is present. The two-dimensional (undamped) wave equation is, in Cartesian coordinates,

$\frac{{\partial }^{2}u}{\partial t^2}=c^2\left(\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}\right)$ (2)

This arises for example when we model the transverse vibrations of a membrane. See Figures 3(a), 3(b). Here $u\left(x,y,t\right)$ is the definition of a point $\left(x,y\right)$ on the membrane at time $t$ . Again, a boundary condition must be specified: commonly

$u=0t\ge 0$

on the boundary of the membrane, if this is fixed (clamped). Also initial conditions must be given

$u\left(x,y,0\right)=f\left(x,y\right)$ (initial position) $\frac{\partial u}{\partial t}\left(x,y,0\right)=g\left(x,y\right)$ (initial velocity)

For a circular membrane, such as a drumhead, polar coordinates defined by $x=rcos\theta ,y=rsin\theta$ would be more convenient than Cartesian. In this case (2) becomes

$\frac{{\partial }^{2}u}{\partial t^2}=c^2\left(\frac{{\partial }^{2}u}{\partial r^2}+\frac{1}{r}\frac{\partial u}{\partial r}+\frac{1}{r^2}\frac{{\partial }^{2}u}{\partial {\theta }^2}\right)0\le r\le R,0\le \theta \le 2\pi$

for a circular membrane of radius $R.$

Figure 3