= Solution
Use downward distance $y=H-z$, with $z$ the height above the bottom, and write $W=W_s(a)>0$. This convention will also reproduce the printed negative deposit-front <velocity> in part (d). The ideal <simple-cubic sphere packing> has <packing fraction> $\pi/6$; leaving $\phi_{\max}$ symbolic keeps the jump calculation independent of its value. In suspension, the <kinematic sedimentation> equation is
$$
\phi_t+\partial_y j(\phi)=0,\qquad j(\phi)=W\phi\left(1-\frac\phi{\phi_{\max}}\right).
$$
Integrate this <conservation law> across a moving discontinuity to obtain the <Rankine-Hugoniot condition>
$$
\boxed{U[\phi]=[j],\qquad U=\frac{j_R-j_L}{\phi_R-\phi_L}.}
$$
The brackets denote values on the increasing-$y$ side minus those on the other side. A <sedimentation shock> is a <concentration> jump moving at this secant slope of the particle flux. Deposited material is stationary and has zero flux.
For the upper clearing front, the states are $0$ and $\phi_{\max}/8$, giving $U_1=7W/8$ downward. For the lower deposition front, the states are $\phi_{\max}/8$ and $\phi_{\max}$, giving $U_2=-W/8$, upward. Thus
$$
\boxed{U_1=\frac78W,\qquad U_2=-\frac18W\quad\text{in downward coordinates}.}
$$
In terms of height above the bottom, the shock paths are
$$
z_{\rm clear}=H-\frac78Wt,\qquad z_{\rm bed}=\frac18Wt.
$$
They meet when $H=Wt$. Therefore \b[complete settling occurs at]
$$
\boxed{t_c=\frac HW,\qquad h_c=\frac H8.}
$$
The final height also follows directly from <particle volume fraction> conservation, $\phi_{\max}h_c=(\phi_{\max}/8)H$. The stationary deposit after the meeting carries no particle flux.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2015/iii/paper-74-monodisperse-shocks.png]
{title=Monodisperse batch sedimentation: clearing and deposition shock paths meet at time H/Ws and height H/8}
The diagram plots time vertically against height, as requested. Its shock slopes have the opposite spatial sign to $U_1,U_2$, because those velocities were defined in the downward coordinate.
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