= Solution
Use the <Burgers reduction of weakly nonlinear rightgoing acoustics>, with $\theta=t-x$, $X=Mx$, $\partial_t=\partial_\theta$ and $\partial_x=-\partial_\theta+M\partial_X$. The <equation of state> expands as
$$
p=\rho+\frac M2(\gamma-1)\rho^2+O(M^2),\qquad
\rho_1=p_1-\frac{\gamma-1}{2}p_0^2.
$$
At order one, <mass conservation> and <momentum conservation> give $(\rho_0-u_0)_\theta=0$ and $(u_0-p_0)_\theta=0$. For the pure rightgoing <acoustic wave> with no independent leading mean density or velocity shift, $\rho_0=u_0=p_0$. This is the background choice implicit in the requested traveling-wave expansion; additional mean components would modify its phase convention. The source's duplicated phrase about expanding $p$ should refer to expanding $\rho$ as well as $u$.
At order $M$, the <mass conservation> equation becomes
$$
(\rho_1-u_1)_\theta+p_{0,X}-(p_0^2)_\theta=0,
$$
whereas the <momentum conservation> equation becomes
$$
(u_1-p_1)_\theta+p_{0,X}=0.
$$
In the latter equation, the two nonlinear products $\rho_0u_{0,\theta}$ and $-u_0u_{0,\theta}$ cancel. Substitute the <equation of state> relation into the former equation and add the two equations to eliminate the first corrections. The result is
$$
\boxed{p_{0,X}-\frac{\gamma+1}{2}p_0p_{0,\theta}=0}.
$$
The same calculation gives $u_1-p_1=-(\gamma+1)p_0^2/4+C(X)$, so the first corrections may be chosen without an explicit secular factor of $\theta$. This is why the retarded-time description can remain uniform for $|\theta|=O(M^{-1})$, provided the leading profile and its relevant <derivatives> remain bounded over that range and the slow propagation interval.
This <Inviscid Burgers equation> can develop a <shock>, so the source's uniformity assertion needs a pre-shock qualification. If $p_0(\theta,0)=F(\theta)$ and $a=(\gamma+1)/2$, its <method of characteristics> gives
$$
p_0=F(s),\qquad \theta=s-aF(s)X,\qquad
p_{0,\theta}=\frac{F'(s)}{1-aXF'(s)}.
$$
The smooth construction ends when this denominator vanishes; if $\max F'>0$, the first such distance is $X_s=[a\max F']^{-1}$.
For viscosity write $\beta=M\nu$ with bounded positive $\nu$. Since $u_{0,xx}=p_{0,\theta\theta}+O(M)$, the order-$M$ <momentum conservation> equation instead reads $(u_1-p_1)_\theta+p_{0,X}=\nu p_{0,\theta\theta}$. Addition now yields the <viscous Burgers equation>
$$
\boxed{p_{0,X}-\frac{\gamma+1}{2}p_0p_{0,\theta}
=\frac\nu2p_{0,\theta\theta}=\frac\beta{2M}p_{0,\theta\theta}}.
$$
Its positive <diffusion> resolves the inviscid steepening rather than supporting an unqualified smooth inviscid expansion through <shock> formation.
Back to article page
