= Solution
The channel is prismatic: its width depends on height, not on $x$. A layer of depth $h$ has cross-sectional area $A=\int_0^h\beta z\,dz=\beta h^2/2$. <Volume conservation> gives $A_t+(Au)_x=0$. With $g'=G\phi$, the integrated excess <hydrostatic pressure> force is
$$
\frac1{\rho_0}\int_0^h p_{\rm excess}\,b(z)\,dz
=G\phi\int_0^h(h-z)\beta z\,dz=\frac{G\phi\beta h^3}{6}.
$$
Thus the streamwise <momentum> balance is $(Au)_t+(Au^2+G\phi\beta h^3/6)_x=0$. Under the specified vertical-settling approximation, the horizontal projection of the depositional boundary has width $\beta h$, so the particle balance is $(A\phi)_t+(Au\phi)_x=-\beta h\,w\phi$, where $w=w_0f(\phi)>0$ is the downward speed magnitude. The <prismatic triangular-channel shallow water equations> are therefore
$$
\boxed{\begin{aligned}
h_t+uh_x+\frac h2u_x&=0,\\
u_t+uu_x+G\phi h_x+\frac{Gh}{3}\phi_x&=0,\\
\phi_t+u\phi_x&=-\frac{2w\phi}{h}.
\end{aligned}}
$$
The pressure coefficient $h/3$ is the section-weighted mean of $h-z$. The factor two in deposition comes from top width divided by area. There is no streamwise widening term, because $\beta$ is constant along the channel.
Define the positive <characteristic speed> scale $c=\sqrt{G\phi h/2}$. In variables $(u,\phi,c)$ the equations become
$$
\begin{aligned}
u_t+uu_x+4cc_x-\frac{4c^2}{3\phi}\phi_x&=0,\\
\phi_t+u\phi_x&=-\frac{2w\phi}{h},\\
c_t+uc_x+\frac c4u_x&=-\frac{wc}{h}.
\end{aligned}
$$
The coefficient matrix of this <hyperbolic system> has <eigenvalues> $u,u+c,u-c$. Hence \b[the three characteristic families are]
$$
\boxed{\frac{dx}{dt}=u,\qquad \frac{dx}{dt}=u\pm c.}
$$
Along $dx/dt=u$, the <concentration> equation is the ordinary differential equation $d\phi/dt=-2w\phi/h$. Along $dx/dt=u\pm c$, the left eigenvectors $(1,\mp4c/(3\phi),\pm4)$ give the <sedimenting triangular-channel characteristic compatibility> equations
$$
\boxed{\frac{du}{dt}\pm4\frac{dc}{dt}\mp\frac{4c}{3\phi}\frac{d\phi}{dt}=\mp\frac{4wc}{3h}.}
$$
Here every derivative in a given equation follows that characteristic family, and $h=2c^2/(G\phi)$. These three compatibility equations form the characteristic description; they are not three independent closed equations for all fields on any one curve. In particular, $u\pm4c$ are not conserved <Riemann invariants> when the <concentration> varies: its differential and the deposition source must both be retained. The description assumes $h,\phi>0$; the dry or zero-buoyancy limit is degenerate.
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