= Area transformation of a nonlinear acoustic simple wave
{title2=$q=gQ,\quad g=\sqrt{A(Z_0)/A(Z)},\quad \zeta=\int_{Z_0}^Zg(s)\,ds$}
For positive smooth tube area $A$, the equation $q_Z-qq_\theta+q(\log A)'/2=0$ becomes $Q_\zeta-QQ_\theta=0$ under the displayed transformation. Indeed, $g'/g=-A'/(2A)$ cancels geometrical spreading and $\zeta'=g$ makes the remaining nonlinear and transport coefficients equal. Starting with $q(\theta,Z_0)=f(\theta)$ gives $Q=f(\xi)$ and $\theta=\xi-f(\xi)\zeta$; <characteristic crossing> first occurs when $\zeta\sup f'=1$, if that supremum is positive.
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