7.1 The Wave Equation

7.1.1 Mathematical Description of a Wave

A wave is a periodic variation or disturbance which travels at a well defined speed through space. How do we describe waves mathematically? Suppose $f(\chi)$ is some periodic function which takes a phase $\chi$ measured in radians (fractions of $2\pi$ ). Then we can describe the variation in space at a specific point in time as a snapshot:

y(x)=Af\left(\frac{2\pi}{\lambda}x+\delta\right),

(7.1)

where $\lambda$ is the wavelength of the wave (the spatial period). We can also describe the variation in amplitude at a single point in space over time:

y(t)=Af\left(\frac{2\pi}{T}t+\theta\right),

(7.2)

where $T$ is the temporal period.

To put these pictures together, we can consider the snapshot picture with a shift $x-vt$ where $v$ is the speed of the wave. Then we have

$\displaystyle y(x,t)$	$\displaystyle=Af\left(\frac{2\pi}{\lambda}(x-vt)+\phi\right)$	(7.3)
	$\displaystyle=Af\left(\frac{2\pi}{\lambda}x-\frac{2\pi v}{\lambda}t+\phi\right)$	(7.4)
	$\displaystyle=Af\left(\frac{2\pi}{\lambda}x-\frac{2\pi}{T}t+\phi\right)$	(7.5)
	$\displaystyle=Af(kx-\omega t+\phi).$	(7.6)

Where we have defined the wavenumber (spatial frequency measured in radians/m) $k=2\pi/\lambda$ and recalled $v=f\lambda$ , $f=1/T$ and $\omega=2\pi f$ .

Example 7.1.

Suppose we have a sinusoidal wave given by $y(x,t)=A\cos(kx-\omega t+\phi)$ . What is the particle velocity at fixed position $x$ ?

To solve this, we take the partial derivative with respect to time (to keep $x$ constant)

\frac{\partial y(x,t)}{\partial t}=A\omega\sin(kx-\omega t+\phi).

(7.7)

Note that this is different to the propagation speed of the wave itself, which is constant.

7.1.2 Constructing the Wave Equation

For simplicity, let’s consider a general right-travelling wave $f(x-vt)$ . We can make a substitution $u=x-vt$ . Then we get

$\displaystyle\frac{\partial f}{\partial x}$	$\displaystyle=\frac{\partial f}{\partial u}\frac{\partial u}{\partial x}$	(7.8)
	$\displaystyle=\frac{\partial f}{\partial u}$	(7.9)
$\displaystyle\implies\frac{\partial^{2}f}{\partial x^{2}}$	$\displaystyle=\frac{\partial^{2}f}{\partial u^{2}},$	(7.10)
$\displaystyle\frac{\partial f}{\partial t}$	$\displaystyle=\frac{\partial f}{\partial u}\frac{\partial u}{\partial t}$	(7.11)
	$\displaystyle=-v\frac{\partial f}{\partial u}$	(7.12)
$\displaystyle\implies\frac{\partial^{2}f}{\partial t^{2}}$	$\displaystyle=v^{2}\frac{\partial^{2}f}{\partial u^{2}}.$	(7.13)

If we do this same calculation with a left-travelling wave $f(x+vt)$ , we get the same relation. Thus, by construction, the general solution to the differential equation

\frac{\partial^{2}f}{\partial t^{2}}=v^{2}\frac{\partial^{2}f}{\partial x^{2}},

(7.14)

is $f(x,t)=f_{l}(x+vt)+f_{r}(x-vt)$ . This linear differential equation is known as the wave equation. All functions which satisfy our definition of a wave solve this equation.