Waves

The displacement, $y(x,t)$, of an elastic string under tension, $T$, with a mass per unit length, $m$, (or equivalently density) satisfies a wave equation

$${\partial^{2}y\over \partial t^{2}} = C^{2}{\partial^{2}y\over \partial x^{2}},$$ (6.1)

where $C^{2} = T/m$. The general solution for an infitinte string is given by d'Alembert's solution

$$f = f(x-Ct) + g(x+Ct),$$ (6.2)

where $f$ and $g$ are arbitrary functions, representing arbitrary wave shapes travelling with speeds $C$ and $-C$ without change of shape. Such waves may be decomposed into Fourier components of the form

$$y = e^{i(kx -\omega t)}.$$ (6.3)

Writing the solution in this complex variable form is just a short hand notation for the general solution

$$y = A\cos (kx - \omega t) + B\sin (kx - \omega t),$$

where $A$ and $B$ are constants. Note that we may rewrite

$$kx - \omega t = k\left ( x - {\omega \over k}t\right ).$$

Whenever we have constant coefficients in the linear wave equation we can look for solutions of the form (6.3). Then substituting into (6.1) we replace

$${\partial \over \partial x} \hbox{ by } ik \hbox{ and } {\partial \over \partial t} \hbox{ by } - i \omega,$$

to obtain

$$\omega^{2} = k^{2}C^{2}.$$ (6.4)

(6.4) is called the dispersion relation, which gives the frequency, $\omega$ of a Fourier component in terms of its wavenumber, $k$ and the wave speed. The phase speed is

$${\omega \over k} = \pm C.$$

The period of the wave is $2\pi /\omega$ and its wavelength is $2\pi / k$.