 Solution

The undisturbed reservoir remains at $h=H$, $u=0$, while the particle budget gives $\varphi ={\varphi }_0{e}^{-V_{s}t/H}$. The edge of the rarefaction wave advancing into that reservoir travels with the local left-running characteristic speed, $-\sqrt{b_{0}H}{e}^{-V_{s}t/(2H)}$. ThusThis is the disturbance head into the initially resting reservoir, also called the leading edge of the rarefaction wave; it reduces to $-c_{0}t$ as $V_s\to 0$. 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 ${\varphi }_0$. The locally uniform shelf approximation gives $b_f=b_0{e}^{-V_{s}t/h_f}$, soThis tends to $u_{f}t$ 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.