= Solution
The undisturbed reservoir remains at $h=H$, $u=0$, while the particle budget gives $\phi=\phi_0e^{-V_st/H}$. The edge of the <rarefaction wave> advancing into that reservoir travels with the local left-running <characteristic speed>, $-\sqrt{b_0H}\,e^{-V_st/(2H)}$. Thus
$$
\boxed{x_r(t)=-\frac{2Hc_0}{V_s}\left[1-e^{-V_st/(2H)}\right]}.
$$
This is the disturbance head into the initially resting reservoir, also called the leading edge of the <rarefaction wave>; it reduces to $-c_0t$ as $V_s\to0$. The rightmost fan/shelf junction is the rarefaction tail, not this reservoir edge. Its trajectory in the full sedimenting system cannot be obtained by retaining the constant-buoyancy <Riemann invariants>.
For the requested near-front approximation, hold $h_f=4H/(F+2)^2$ fixed and take the shelf's concentration initially to be $\phi_0$. The locally uniform shelf approximation gives $b_f=b_0e^{-V_st/h_f}$, so
$$
\dot x_f=F\sqrt{b_0h_f}\,e^{-V_st/(2h_f)},\qquad \boxed{x_f(t)\simeq\frac{2Fh_f\sqrt{b_0h_f}}{V_s}\left[1-e^{-V_st/(2h_f)}\right]}.
$$
This tends to $u_ft$ at small settling and predicts a finite limiting front location within the fixed-depth closure. It is not an exact solution of the spatially varying <variable-buoyancy shallow water equations>: different material histories and resulting buoyancy gradients eventually invalidate a perfectly uniform shelf. No exact finite-time stoppage is claimed.
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