Area transformation of a nonlinear acoustic simple wave ($q=gQ,g=\sqrt{A(Z_0)/A(Z)},\zeta ={\int }_{Z_0}^{Z}g(s)ds$)

For positive smooth tube area $A$, the equation $q_Z-q{q}_{\theta }+q(\log A{)}^{\prime }/2=0$ becomes $Q_{\zeta }-Q{Q}_{\theta }=0$ under the displayed transformation. Indeed, $g^{\prime }/g=-A^{\prime }/(2A)$ cancels geometrical spreading and ${\zeta }^{\prime }=g$ 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 $\zeta \sup f^{\prime }=1$, if that supremum is positive.

For area $A(R)=R^2$ and smooth data $q(\theta ,R_0)=f(\theta )$ at $R_0>0$, the transformed distance is $\zeta =R_0\log (R/R_0)$. If $M=\sup f^{\prime }>0$, the first shock wave occurs at the displayed radius; if $M\le 0$, there is no forward characteristic crossing. Nontrivial bounded data cannot instead be imposed at $R=0$ in this outgoing simple-wave model: along every characteristic, $Rq$ is constant, so a bounded limit at the origin forces that constant to vanish. With distance measured from a finite emitting surface, use $A(Z)=(R_0+Z{)}^2$ and subtract $R_0$ from the shock radius.