Sound Waves

There are many different types of wave motions in physical situations. Sound waves occur in fluids and non-magnetic plasmas and arise when there are variations in the pressure. Consider a fluid or plasma satisfying, the equation of motion

$$\rho \left ({\partial {\bf v}\over \partial t} + ({\bf v}\cdot \nabla){\bf v} \right ) = - \nabla p,$$ (6.5)

the equation of mass continuity,

$${\partial \rho \over \partial t} + \nabla \cdot \left (\rho {\bf v}\right ) = 0,$$ (6.6)

and an equation of state which we take as the gas law

$$p = R\rho T,$$ (6.7)

where the gas constant, $R$, and the temperature, $T$, are taken as constants and the fluid is isothermal.

Now the basic situation is regarded as a uniform medium at rest ( ${\bf v} = {\bf0}$) with a uniform pressure ($p_{0}$) and uniform density ($\rho_{0}$) satisfying

$$p_{0} = R \rho_{0}T.$$

Note that the other equations, (6.5) and (6.6) are satisfied automatically. We assume that there is a small change in pressure, $p_{1}$, which introduces a small velocity, ${\bf v_{1}}$, and causes small change in the density, $\rho_{1}$. These perturbations

$$p_{1}({\bf r}, t), \qquad {\bf v}_{1}({\bf r}, t), \qquad \rho_{1}({\bf r}, t),$$

are small in magnitude and much smaller than the magnitude of the initial uniform medium values of $p_{0}$ and $\rho_{0}$. Hence,

$$p_{1} \ll p_{0}, \qquad \rho_{1} \ll \rho_{0}.$$

Thus, we can write the actual density as

$$\rho = \rho_{0} + \rho_{1},$$

and similarly for the pressure. Hence, (6.5), (6.6) and (6.7) are expressed as

$$(\rho_{0} + \rho_{1})\left [{\partial {\bf v}_{1}\over \par... ...cdot \nabla ){\bf v}_{1}\right ] = - \nabla (p_{0} + p_{1}),$$

$${\partial \over \partial t}\left (\rho_{0} + \rho_{1}\right... ...bla \cdot \left ((\rho_{0} + \rho_{1}){\bf v}_{1}\right ) = 0,$$

and

$$p_{0} + p_{1} = RT(\rho_{0} + \rho_{1}).$$

Now we use the fact that $p_{0}$ and $\rho_{0}$ are uniform, neglect all products of small quantities to give

$$\rho_{0}{\partial {\bf v}_{1}\over \partial t} = - \nabla p_{1},$$ (6.8)

$${\partial \rho_{1}\over \partial t} + \rho_{0}\nabla \cdot {\bf v}_{1} = 0,$$

$$p_{1} = c_{s}^{2}\rho_{1},$$

where $c_{s}^{2} = RT$ is the square of the isothermal sound speed (typical values for the sound speed are 300m/s for air, 270m/s for hydrogen, 1410m/s for water and 5600m/s for glass). Notice that these equations are linear equations in the perturbed quantities and that they have constant coefficients.

We can eliminate the velocity ${\bf v}_{1}$ and $p_{1}$ from the equations to derive a single equation for the density perturbations, $\rho_{1}$. Hence,

$${\partial^{2}\rho_{1}\over \partial t^{2}} = c_{s}^{2}\nabla^{2}\rho_{1}.$$ (6.9)

This is a general wave equation. If we consider a 1-D problem, so that $\rho_{1}(x,t)$, then

$${\partial^{2}\rho_{1}\over \partial t^{2}} = c_{s}^{2}{\partial^{2}\rho_{1}\over \partial x^{2}},$$

and we have the usual wave equation. Solutions to this wave equation represent sound waves propagating with speed $c_{s}$ along the $x$-direction. If we write,

$$\rho_{1} = \tilde{\rho}e^{i(kx - \omega t)} \quad \Rightarr... ..._{s}^{2} = \tilde{\rho}e^{i({\bf k}\cdot {\bf r} - \omega t)},$$

and $\tilde{\rho}$ is the constant amplitude of the wave.

More generally, a single wave mode in three dimensions has $\rho_{1}(x,y,z,t)$ and we consider

$$\rho_{1} = \tilde{\rho}e^{i(kx + ly + mz - \omega t)},$$ (6.10)

which represents a wave travelling in the direction of ${\bf k}$

$${\bf k} = \left ( k, l, m\right ),$$

with wavenumber

$$\vert{\bf k}\vert = \sqrt{k^{2} + l^{2} + m^{2}}.$$

Substituting (6.10) into (6.9) gives the dispersion relation

$$\omega^{2} = \left ( k^{2} + l^{2} + m^{2} \right )c_{s}^{2}.$$ (6.11)

Note

1. Sound waves propagate equally in all directions, there is no preferred direction.

2. From (6.8)
   
   $$-i \omega \rho_{0}{\bf v}_{1} = - i{\bf k}p_{1},$$
   
   
   and so ${\bf v}_{1}$ is in the same direction as ${\bf k}$. This means that sound waves propagate longitudinally.