Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 75 / 3 / d

Without settling, $b=b_0=G{\varphi }_0$ and $c_0=\sqrt{b_{0}H}$. The left-running rarefaction wave has the unchanged Riemann invariant $u+2c=2c_0$ and similarity coordinate $\xi =x/t=u-c$. HenceMatching to the Froude number condition $u_f=F{c}_f$ givesThe full outer profiles are $(h,u)=(H,0)$ for $x/t<-c_0$; the displayed fan for $-c_0<x/t<{\xi }_s$; and $(h,u)=(h_f,u_f)$ for ${\xi }_s<x/t<u_f$, whereAhead of $x_f$ there is only ambient fluid, so the particle-laden layer has zero depth. The shelf width is $(u_f-{\xi }_s)t=c_{f}t>0$ for every finite positive $F$. It disappears only in the formal $F\to \infty$ dry-front limit, where $u_f\to 2c_0$. For $F<1$, the fan/shelf junction lies to the left of the initial barrier; the profiles remain consistent. This is the finite-Froude lock-release rarefaction, valid while the reservoir can be treated as semi-infinite.