= Solution
Use the upward coordinate of the preceding <kinematic sedimentation> calculation, with the given smooth profile on $0\le\zeta\le h$ and clear fluid above. This asks for the first characteristic breaking of that profile; a separately imposed packed-bed discontinuity would be a different boundary problem.
Substitution of the initial <concentration> into the <characteristic speed> gives
$$
c_0(\zeta)=-\frac{V_s}{3}-\frac{2V_s\zeta^2}{3h^2},\qquad
z(\zeta,t)=\zeta+c_0(\zeta)t.
$$
The characteristic Jacobian is
$$
\frac{\partial z}{\partial\zeta}=1-\frac{4V_s\zeta t}{3h^2}.
$$
It first vanishes at the upper edge $\zeta=h$. That edge has zero <concentration> and descends initially at $V_s$, so
$$
\boxed{t_s=\frac{3h}{4V_s},\qquad z_s=h-V_st_s=\frac h4.}
$$
Although the derivative of the initial profile jumps to zero in the clear region, the <concentration> itself is continuous there. The <shock> starts with vanishing amplitude. Both limiting concentrations at onset are zero, and the <Rankine-Hugoniot condition> consequently gives
$$
\boxed{\dot s(t_s^+)=-V_s.}
$$
This is a downward speed $V_s$.
After formation the upper state remains clear, $\phi_R=0$, and the lower state is positive. The <parabolic-profile sedimentation shock> has
$$
\dot s=-V_s\left(1-\frac{\phi_L}{\phi_{\max}}\right).
$$
It therefore initially slows down in magnitude: its signed upward-coordinate speed increases from $-V_s$. One can also verify that its captured left state increases. Write $r=\zeta_L/h$, where $\zeta_L$ labels the incident lower characteristic. Conservation of the particle volume initially above that characteristic gives
$$
\frac{V_s t}{\phi_{\max}}\phi_0(\zeta_L)^2=\int_{\zeta_L}^h\phi_0(\zeta)\,d\zeta,
\qquad
\frac{V_st}{h}=\frac{r+2}{(r+1)^2}.
$$
The derivative of the right-hand side with respect to $r$ is $-(r+3)/(r+1)^3<0$. Thus $r$ decreases as time increases, and $\phi_L=\phi_{\max}(1-r^2)/3$ increases. The downward <shock> speed continues to decrease on this branch until a bed or another boundary interacts with it. No later boundary behavior is needed for the requested conclusion.
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