If the self-gravity is ignorable, equation (2.35)
![\begin{displaymath}
\frac{\partial \delta \rho}{\partial t}+\rho_0 \frac{\partial \delta u}{\partial x}=0,
\end{displaymath}](img455.png) |
(2.40) |
and equation (2.37)
![\begin{displaymath}
\rho_0 \frac{\partial \delta u}{\partial t}= -c_{is}^2\frac{\partial \delta \rho}{\partial x},
\end{displaymath}](img456.png) |
(2.41) |
where we assumed the gas is isothermal,
these two equations describe the propagation and growth of perturbations.
If the gas acts adiabatically, replace
with
.
Making
(2.40) and
(2.41)
vanishes
and we obtain
![\begin{displaymath}
\frac{\partial^2 \delta u}{\partial t^2}-c_{is}^2\frac{\partial^2 \delta u}{\partial x^2}=0.
\end{displaymath}](img462.png) |
(2.42) |
Since this leads to
equation (2.42) has a solution that a wave propagates with a phase velocity of
.
Since the displacement (
) is parallel to the propagation direction
,
and the restoring force comes from the pressure, this seems the sound wave.
Subsections
Kohji Tomisaka
2012-10-03