Fourier Analysis of (4.15)

Since the coefficients of (4.15) are constant in both space and time we may look for plain waves and Fourier analyse $\Delta$ by assuming

$$\Delta = \Delta_{0} e^{i(\omega t - {\bf k}\cdot {\bf r})},$$ (4.16)

where $\Delta_{0}$ is a constant and ${\bf k}\cdot {\bf r} = kx + ly + mz$. In (4.16) we have

$$\omega \hbox{ -- the {\it frequency}}$$ (4.17)

$${\bf k} = ( k, l, m) \hbox{ -- the {\it wave vector}}$$ (4.18)

and

$${\bf r} = ( x, y, z) \hbox{ -- the {\it position vector}}$$ (4.19)

This form for $\Delta$ is particularly useful since we note that

$${\partial \over \partial t} = i \omega, \qquad {\partial^{2} \over \partial t^{2}} = - \omega^{2},$$

$${\partial \over \partial x} = -i k, \qquad {\partial^{2} ... ... = - k^{2}, \hbox{ where $k$\ is the horizontal wavenumber},$$

$$\nabla \cdot = - i {\bf k} \cdot, \qquad \nabla = -i {\bf k}, \qquad \nabla \times = - i{\bf k} \times.$$

Thus, defining the total wavenumber as $K^{2} = k^{2} + l^{2} + m^{2}$, (4.15) becomes

$$-\omega^{2}\Delta = - K^{2}c_{s}^{2}\Delta \qquad \Rightarrow \qquad (\omega^{2} - K^{2}c_{s}^{2})\Delta = 0.$$ (4.20)

So either $\Delta = 0$ (and we have the trivial solution since $\rho_{1}=0$, $p_{1} = 0$ and so ${\bf v}_{1} = 0$) or or the coefficient of $\Delta$ in (4.20) must vanish. Hence,

$$\omega^{2} = K^{2}c_{s}^{2}.$$ (4.21)

This is the dispersion relation for sound waves and it relates the frequency with which the waves oscillates in time to the spatial length scales of the wave through the wave vector (and the various wave numbers). The dispersion relation, $\omega = \omega ({\bf k})$, can be used to define two important quantities, namely the phase speed and the group velocity.

The phase speed, in general, is given by

$$c_{ph} = {\omega \over K},$$ (4.22)

and in the case of sound waves this is

$$c_{ph} = {\omega \over K} = \pm c_{s}.$$ (4.23)

Thus, the phase speed is the sound speed.

The group velocity is

$${\bf c}_{g} = {\partial w \over \partial {\bf k}} = \left (... ...ver \partial l}, {\partial \omega \over \partial m}\right ).$$ (4.24)

For sound waves

$$\omega^{2} = \left (k^{2} + l^{2} + m^{2} \right ) c_{s}^{2}.$$

Differentiating this gives

$$2 \omega {\partial \omega \over \partial {\bf k}} = \left (2 k c_{s}^{2}, 2 l c_{s}^{2}, 2 m c_{s}^{2}\right ),$$

$$\Rightarrow \qquad {\bf c}_{g} = {c_{s}^{2}\over \omega }( ... ... c_{s}^{2}{K\over \omega} \hat{\bf k} = \pm c_{s} \hat{\bf k}.$$ (4.25)

on using (4.23).

The phase speed gives the speed of an individual wave and the phase velocity is $c_{ph} \hat{\bf k}$. The group velocity gives the speed and direction of the transport of information and energy.