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Sound Waves - Basic Waves Properties

Consider the simplest wave problem first by setting the magnetic field and gravity to zero. This is equivalent to a high $\beta$ plasma and is valid for lengths that are much shorter than the pressure scale height. Thus, the equilibrium satisfies

\begin{displaymath}
\nabla p_{0} = 0 \qquad \Rightarrow \qquad p_{o} = {\rm constant}.
\end{displaymath}

The equilibrium is uniform. No restriction is placed on $\rho_{0}$, the equilibrium density but, for simplicity, it is also assumed uniform as well.
\begin{displaymath}
p_{0} = {\rm constant}, \qquad \rho_{0} = {\rm constant}.
\end{displaymath} (4.12)

With (4.12), (4.11) reduces to
\begin{displaymath}
\rho_{0}{\partial^{2}{\bf v}_{1}\over \partial t^{2}} = \gamma p_{0}
\nabla \left (\nabla \cdot {\bf v}_{1}\right ).
\end{displaymath} (4.13)

To progress, define $\Delta = \nabla \cdot {\bf v}_{1}$ and take the divergence of (4.11). This gives a simple wave equation for the scalar quantity $\Delta$ of the form

\begin{displaymath}
{\partial^{2}\over \partial t^{2}}\Delta = {\gamma p_{0}\over
\rho_{0}}\nabla ^{2}\Delta .
\end{displaymath}

The characterstic speed is the sound speed, $c_{s}$, where
\begin{displaymath}
c_{s}^{2} = {\gamma p_{0}\over \rho_{0}},
\end{displaymath} (4.14)

and the wave equation for sound waves is
\begin{displaymath}
{\partial^{2}\over \partial t^{2}}\Delta = c_{s}^{2}\nabla ^{2}\Delta .
\end{displaymath} (4.15)

Notice that these waves travel at the speed of sound and assume that $\Delta$ is non-zero. This means that sound waves are compressional.

Subsections
next up previous
Next: Fourier Analysis of (4.15) Up: Magnetohydrodynamic Waves Previous: Linearised MHD Equations
Prof. Alan Hood
2000-01-11