Gas Dynamic to Acoustic wave equation

[Trevorman]

Homework Statement

Derive from the formulas
$\frac{D^\pm}{Dt}(u \pm F) = 0$

where
$\frac{D^\pm}{Dt} = \frac{\partial}{dt} + ( u \pm c) \frac{\partial}{\partial x}$
the one-dimensional wave equation in the acoustical limit.

\begin{cases}
u << c\\
c \approx c0 = const\\
F = \frac{2c}{\gamma-1}
\end{cases}

Homework Equations

The answer should be
$\frac{1}{c_0^2} \frac{\partial^2 p}{\partial t^2} = \frac{\partial^2 p}{\partial x^2}$
where
$c_0 = \sqrt{\frac{\gamma p_0}{\rho_0}}$

The Attempt at a Solution

Expanding
$\frac{D}{Dt} = \frac{\partial}{\partial t}+\frac{\partial u}{\partial x} \pm \frac{\partial c}{\partial x}$

Now combining equations
$\left( \frac{\partial}{\partial t}+\frac{\partial u}{\partial x} \pm\frac{\partial c}{\partial x} \right) \cdot (u \pm F) = 0 \Rightarrow$
$\Rightarrow\dot{u} \pm\dot{F} + uu^\prime + Fu^\prime + uc^\prime +Fc^\prime = 0$

Since $c = c_0 = const$ and $u<<c \Rightarrow uc^\prime \approx 0$, the quadratic terms are neglected. $\dot{F} = 0$ ,and substituting F
$\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + \frac{2c}{\gamma-1} \frac{\partial c }{\partial x} = 0$

This should be done by substituting

$c=c_0 + c^\prime$
$u=u^\prime$
$p = p_0 + p^\prime$
$\rho = \rho_0 + \rho^\prime$
and by neglecting small terms as $u^\prime c^\prime$

I do not know how to proceed with this example.

Thank you!

 

[hunt_mat]

This is nothing but the characteristic form of the 1-D gas equations. No perturbations are necessary.
1) Write out the compressible Euler equations in 1D
2) Note that c^2=\gamma p/rho
3) The flow you're looking for is isotropic so p=A\rho^{\gamma}
4) Play.