= Solution
Put the broken dam at $x=0$ and let $c_0=\sqrt{gH/3}$. A dry bed is assumed downstream. The left-family <rarefaction wave> joins the undisturbed reservoir $h=H,u=0$ to the dry front. The other <Riemann invariant> is fixed at $u+6c=6c_0$. Within the fan its characteristic speed equals $\xi=x/t=u-c$, so
$$
c=\frac{6c_0-\xi}{7},\qquad
u=\frac6{7}(c_0+\xi).
$$
Thus the <dry-bed dam break with cubic wetted area> has the fields
$$
\boxed{h(x,t)=\begin{cases}
H,&x/t\leq-c_0,\\
H\left[\frac{6-x/(c_0t)}7\right]^2,&-c_0<x/t<6c_0,\\
0,&x/t\geq6c_0,
\end{cases}}
$$
$$
\boxed{u(x,t)=\begin{cases}
0,&x/t\leq-c_0,\\
\frac6{7}(c_0+x/t),&-c_0<x/t<6c_0.
\end{cases}\qquad x_f(t)=6c_0t.}
$$
Velocity is not a meaningful fluid quantity in the dry region; its wet-side limiting value is $u_f=6c_0=2\sqrt{3gH}$. The upstream head travels at $-c_0$, while the front travels at the stated constant speed in this inviscid model.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2003/iii/paper-76-dam-break.png]
{title=Depth and velocity profiles for the dry-bed dam break with the printed cubic wetted-area geometry}
Bottom drag slows the flow and changes the dry-tip structure, where the depth is small and bed stress per unit fluid mass becomes especially important. It can replace the inviscid zero-depth high-speed tail by a drag-controlled nose. An exact altered profile or front speed requires a specified drag law; the inviscid speed should not be extrapolated to a thin real irrigation-channel front without that qualification.
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