= Parabolic-profile sedimentation shock
For $\phi_0(z)=\phi_{\max}(1-z^2/h^2)/3$ on $0\le z\le h$, with clear fluid above and no imposed bed discontinuity, the first compression of the smooth profile occurs at $t=3h/(4V_s)$ and $z=h/4$. The nascent shock enters clear fluid with initial signed speed $-V_s$. Its downward speed subsequently decreases as the state behind it becomes more concentrated, until a boundary interaction changes the problem. This is characteristic breaking, not the time of initiation of a separately imposed packed-bed front.
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