= Solution
Take $z$ upward and $u_s>0$ downward. The <quadratic hindered-settling flux> is $j(\phi)=-u_s\phi(1-\phi)$. Particle <mass conservation> gives the <kinematic sedimentation> equation
$$
\phi_t+j(\phi)_z=0,\qquad \frac{dz}{dt}=j'(\phi)=u_s(2\phi-1),\qquad\frac{d\phi}{dt}=0
$$
on a smooth <characteristic curve>. Integrate the <scalar conservation law> across a moving discontinuity: $V[\phi]=[j]$. The <Rankine-Hugoniot condition> gives the <settling shock speed>
$$
\boxed{V=\frac{j(\phi_R)-j(\phi_L)}{\phi_R-\phi_L}=u_s(\phi_L+\phi_R-1),}
$$
where $L$ is the lower side and $R$ the upper side. Since $j''=2u_s>0$, a compressive <shock> has $\phi_L>\phi_R$ and its neighboring <characteristic curves> point into it.
Put $a=\phi_1$, so $\phi_2=2a$, with $0<a<1/4$. The <two-layer sedimentation with a compression shock> has
$$
\boxed{V_1=-u_s(1-a),\qquad V_2=-u_s(1-3a),\qquad V_3=2u_sa.}
$$
These respectively separate upper suspension from clear fluid, lower suspension from upper suspension, and deposit from lower suspension. Their initial positions are $H,z_m,0$. The <characteristic curves> in the two suspended states have slopes $u_s(2a-1)$ and $u_s(4a-1)$; clear-fluid and deposit <characteristic curves> have slopes $-u_s$ and $u_s$.
For $z_m=H/2$, $V_2$ and $V_3$ meet first. Their intersection gives
$$
\boxed{t_{c1}=\frac{z_m}{u_s(1-a)}=\frac{H}{2u_s(1-a)},\qquad h_{c1}=\frac{2az_m}{1-a}=\frac{aH}{1-a}.}
$$
The potential meeting of the upper two fronts would take $(H-z_m)/(2u_sa)$, which is later in the stated range. The new front separates <particle volume fraction> one from <particle volume fraction> $a$, so
$$
\boxed{V_4=u_sa.}
$$
It follows $z=h_{c1}+u_sa(t-t_{c1})$ until it meets $z=H-u_s(1-a)t$. The final mass-conserving intersection is
$$
\boxed{t_{c2}=\frac{H(1-a)-az_m}{u_s(1-a)},\qquad h_{c2}=a(H+z_m).}
$$
For $z_m=H/2$, these are $H(1-3a/2)/[u_s(1-a)]$ and $3aH/2$.
The PDF prints a plus sign before $az_m/H$ in the stopping-time numerator. That sign contradicts both the top-front trajectory and particle <mass conservation>; the correct sign is minus, as derived here. Once all particles are deposited, the remaining $1$-to-$0$ front is stationary because both limiting fluxes vanish.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2016/iii/paper-330-compression-shocks.png]
{title=Settling compression shocks, their merger, and the triple-point initial layering}
The first panel shows the two-front merger and the final intersection, with straight <characteristic curves> in each suspended layer. The second panel shows the tuned three-front intersection.
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