= Solution
The <shallow-water approximation> requires a small depth-to-horizontal-scale ratio, slow streamwise variation of the channel and interface, small vertical acceleration, and rapid enough cross-sectional mixing for $u$ and $\phi$ to be representative averages. Near the channel apex, or during the initial collapse at a sharp front, these conditions need not hold. The high <Reynolds number> permits neglect of leading viscous resistance; weak stirring mixes the particles without appreciable ambient <fluid entrainment>.
Put $\lambda=\beta/H$. At elevation $z$, the channel width is $2\lambda xz$, so the occupied area and interface width are
$$
A=\lambda xh^2,\qquad T=2\lambda xh.
$$
<Volume conservation> and suspended-particle conservation give
$$
A_t+(Au)_x=0,\qquad (A\phi)_t+(Au\phi)_x=-W_sT\phi.
$$
The excess <hydrostatic pressure> is $\rho_0d\phi(h-z)$. Its streamwise gradient averaged over the triangular section gives acceleration $-d\phi h_x-(dh/3)\phi_x$: the weighted mean of $h-z$ is $h/3$. Thus the <triangular-channel shallow water equations> are
$$
\boxed{
\begin{aligned}
h_t+uh_x+\frac h2u_x&=-\frac{uh}{2x},\\
u_t+uu_x+d\phi h_x+\frac{dh}{3}\phi_x&=0,\\
\phi_t+u\phi_x&=-\frac{2W_s\phi}{h}.
\end{aligned}}
$$
The quasilinear coefficient <matrix> in $(h,u,\phi)$ is
$$
\begin{pmatrix}u&h/2&0\\d\phi&u&dh/3\\0&0&u\end{pmatrix}.
$$
Its three distinct real <eigenvalues>, for $h,d\phi>0$, establish strict <hyperbolicity>:
$$
\boxed{\frac{dx}{dt}=u,\qquad\frac{dx}{dt}=u\pm c,\qquad c=\sqrt{\frac{d\phi h}{2}}.}
$$
On the material <characteristic curve>, $d\phi/dt=-2W_s\phi/h$. On each of the other two <characteristic curves>, multiplying by the left <eigenvector> $(\pm2c/h,1,\pm dh/(3c))$ gives the compatibility <ordinary differential equation>
$$
\boxed{\frac{du}{dt}\pm\frac{2c}{h}\frac{dh}{dt}\pm\frac{h}{3c}\frac{d g'}{dt}
=\mp c\left(\frac ux+\frac{4W_s}{3h}\right),\qquad \frac{dx}{dt}=u\pm c.}
$$
The derivatives are along the indicated characteristic, with $g'=d\phi$. At vanishing concentration, the gravity-wave speeds coalesce and strict <hyperbolicity> is lost.
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