= Solution
{c}
For one-dimensional <inviscid flow>, <conservation of mass> and the <Euler equation> are
$$
\partial_t\rho+\partial_x(\rho u)=0,\qquad
\partial_tu+u\partial_xu=-\rho^{-1}\partial_xp.
$$
Substitute $u=U_0+v_1$ and the stated background-plus-perturbation expressions. Dropping products of perturbations gives the <linearization>
$$
D_0\rho_1+\rho_0\partial_xv_1=0,\qquad
\rho_0D_0v_1=-\partial_xp_1,\qquad D_0=\partial_t+U_0\partial_x.
$$
The <Taylor expansion> of the <equation of state> gives
$$
p_0\left(1+\frac{\rho_1}{\rho_0}\right)^\gamma
=p_0+\frac{\gamma p_0}{\rho_0}\rho_1+O(\rho_1^2),\qquad
\boxed{p_1=c_0^2\rho_1,\quad c_0^2=\left.\frac{dp}{d\rho}\right|_{\rho_0}=\frac{\gamma p_0}{\rho_0}.}
$$
Since the background coefficients are constant, $D_0$ commutes with $\partial_x$. Applying $D_0$ to the linearized <continuity equation> and using the linearized <Euler equation> therefore yields
$$
D_0^2\rho_1=-\rho_0\partial_xD_0v_1=\partial_x^2p_1,
\qquad
\boxed{(\partial_t+U_0\partial_x)^2\rho_1=c_0^2\partial_x^2\rho_1.}
$$
This is the <convected acoustic wave equation>: in the coordinate $x-U_0t$, it becomes the ordinary <one-dimensional wave equation>. Positive $c_0^2$ is required for propagating <sound waves>.
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