 Solution

The channel is prismatic: its width depends on height, not on $x$. A layer of depth $h$ has cross-sectional area $A={\int }_0^h\beta zdz=\beta h^2/2$. Volume conservation gives $A_t+(Au{)}_x=0$. With $g^{\prime }=G\varphi$, the integrated excess hydrostatic pressure force isThus the streamwise momentum balance is $(Au{)}_t+(A{u}^2+G\varphi \beta h^3/6{)}_x=0$. Under the specified vertical-settling approximation, the horizontal projection of the depositional boundary has width $\beta h$, so the particle balance is $(A\varphi {)}_t+(Au\varphi {)}_x=-\beta hw\varphi$, where $w=w_{0}f(\varphi )>0$ is the downward speed magnitude. The prismatic triangular-channel shallow water equations are thereforeThe 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 $c=\sqrt{G\varphi h/2}$. In variables $(u,\varphi ,c)$ the equations becomeThe coefficient matrix of this hyperbolic system has eigenvalues $u,u+c,u-c$. Hence the three characteristic families areAlong $dx/dt=u$, the concentration equation is the ordinary differential equation $d\varphi /dt=-2w\varphi /h$. Along $dx/dt=u\pm c$, the left eigenvectors $(1,\mp 4c/(3\varphi ),\pm 4)$ give the sedimenting triangular-channel characteristic compatibility equationsHere every derivative in a given equation follows that characteristic family, and $h=2c^2/(G\varphi )$. These three compatibility equations form the characteristic description; they are not three independent closed equations for all fields on any one curve. In particular, $u\pm 4c$ are not conserved Riemann invariants when the concentration varies: its differential and the deposition source must both be retained. The description assumes $h,\varphi >0$; the dry or zero-buoyancy limit is degenerate.