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Next: Electromagnetic Waves Up: Waves Previous: 2D Vibrations of a

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

the equation of mass continuity,
\begin{displaymath}
{\partial \rho \over \partial t} + \nabla \cdot \left (\rho {\bf v}\right ) = 0,
\end{displaymath} (6.6)

and an equation of state which we take as the gas law
\begin{displaymath}
p = R\rho T,
\end{displaymath} (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

\begin{displaymath}
p_{0} = R \rho_{0}T.
\end{displaymath}

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

\begin{displaymath}
p_{1}({\bf r}, t), \qquad {\bf v}_{1}({\bf r}, t), \qquad \rho_{1}({\bf r}, t),
\end{displaymath}

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

\begin{displaymath}
p_{1} \ll p_{0}, \qquad \rho_{1} \ll \rho_{0}.
\end{displaymath}

Thus, we can write the actual density as

\begin{displaymath}
\rho = \rho_{0} + \rho_{1},
\end{displaymath}

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

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


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

and

\begin{displaymath}
p_{0} + p_{1} = RT(\rho_{0} + \rho_{1}).
\end{displaymath}

Now we use the fact that $p_{0}$ and $\rho_{0}$ are uniform, neglect all products of small quantities to give
\begin{displaymath}
\rho_{0}{\partial {\bf v}_{1}\over \partial t} = - \nabla p_{1},
\end{displaymath} (6.8)


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


\begin{displaymath}
p_{1} = c_{s}^{2}\rho_{1},
\end{displaymath}

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,

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

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

\begin{displaymath}
{\partial^{2}\rho_{1}\over \partial t^{2}} =
c_{s}^{2}{\partial^{2}\rho_{1}\over \partial x^{2}},
\end{displaymath}

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,

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

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

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

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

\begin{displaymath}
{\bf k} = \left ( k, l, m\right ),
\end{displaymath}

with wavenumber

\begin{displaymath}
\vert{\bf k}\vert = \sqrt{k^{2} + l^{2} + m^{2}}.
\end{displaymath}

Substituting (6.10) into (6.9) gives the dispersion relation
\begin{displaymath}
\omega^{2} = \left ( k^{2} + l^{2} + m^{2} \right )c_{s}^{2}.
\end{displaymath} (6.11)

Note

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

  2. From (6.8)

    \begin{displaymath}
-i \omega \rho_{0}{\bf v}_{1} = - i{\bf k}p_{1},
\end{displaymath}

    and so ${\bf v}_{1}$ is in the same direction as ${\bf k}$. This means that sound waves propagate longitudinally.

next up previous
Next: Electromagnetic Waves Up: Waves Previous: 2D Vibrations of a
Prof. Alan Hood
2000-11-06