= Shock distance for a spherical simple wave launched at finite radius
{title2=$R_s=R_0\exp[1/(R_0\sup f')]$}
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 $M=\sup f'>0$, the first <shock wave> occurs at the displayed radius; if $M\leq0$, 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.
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