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Next: Alfvén Waves Up: Sound Waves - Basic Previous: Fourier Analysis of (4.15)

Alternative approach to sound waves

We can derive the dispersion relation for sound waves direct from the linearised MHD equations, (4.6) - (4.9) (with ${\bf B}_{0} = 0, {\bf g} = 0$), using the Fourier representation. This, of course can be done because the equilibrium is uniform.
\begin{displaymath}
% latex2html id marker 384(\ref{lincon}) \qquad \Rightarr...
...\over
\rho_{0}} = {({\bf k}\cdot {\bf v}_{1})\over \omega }.
\end{displaymath} (4.26)


\begin{displaymath}
% latex2html id marker 398(\ref{linmom}) \qquad \Rightarr...
...row \qquad {\bf v}_{1} =
{p_{1}\over \omega \rho_{0}}{\bf k}
\end{displaymath} (4.27)


\begin{displaymath}
% latex2html id marker 412(\ref{linenergy}) \Rightarrow p...
...er \rho_{0}} \rho_{1} \Rightarrow
p_{1} = c_{s}^{2}\rho_{1}.
\end{displaymath} (4.28)

From these equations we can see that Now take the scalar product of (4.27) with ${\bf k}$ to get

\begin{displaymath}
{\bf k}\cdot {\bf v}_{1} = {p_{1}\over \omega \rho_{0}}K^{2},
\end{displaymath}

and then using (4.26) and (4.28) we get

\begin{displaymath}
{\bf k}\cdot {\bf v}_{1} = \omega {\rho_{1}\over \rho_{0}} ...
...\over \rho_{0}} = {K^{2}\over \omega
}{p_{1}\over \rho_{0}}.
\end{displaymath}

Therefore,

\begin{displaymath}
\omega^{2} = K^{2}c_{s}^{2}.
\end{displaymath}

as before.
next up previous
Next: Alfvén Waves Up: Sound Waves - Basic Previous: Fourier Analysis of (4.15)
Prof. Alan Hood
2000-01-11