= Burgers reduction of weakly nonlinear rightgoing acoustics
{c}
{title2=$p_X-\tfrac{\gamma+1}{2}p\,p_\theta=\tfrac{\beta}{2M}p_{\theta\theta}$}
For a pure rightgoing <acoustic wave>, retarded time $\theta=t-x$ and slow distance $X=Mx$ separate fast oscillations from nonlinear steepening. Expanding the <equation of state> gives $p=\rho+M(\gamma-1)\rho^2/2+O(M^2)$. The leading perturbations satisfy $u_0=\rho_0=p_0$. Adding the first-order <mass conservation> and <momentum conservation> equations cancels the unknown first corrections and yields the displayed <viscous Burgers equation>. Inviscid characteristics satisfy $\theta=s-(\gamma+1)F(s)X/2$, $p=F(s)$; loss of invertibility marks <shock> formation and the end of the smooth inviscid approximation.
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