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Assume the ambient fluid has zero downslope velocity, no bed drag, and an entrainment coefficient $\alpha >0$. The entraining shallow-water layer receives volume at rate $\alpha |u|$, but neither excess buoyancy nor downslope momentum from that ambient fluid. Consequently volume conservation, conservation of excess buoyancy, and momentum conservation giveThe pressure contribution is the integral of the hydrostatic pressure from part (a). The last source is the downslope component of the gravitational force.

With $D={\partial }_t+u{\partial }_x$, the equivalent equations areThe last term is the momentum required to accelerate newly entrained fluid. For variables $(h,u,b)$, the matrix of the spatial derivatives isThus it is a strictly hyperbolic system for $h,b>0$ and $0\le \theta <\pi /2$. At $\theta =\pi /2$ all three characteristic speeds coincide and the matrix is not diagonalizable. The exclusion of that angle presumes the physical range with $\cos \theta \ge 0$: allowing an overhanging channel with $\cos \theta <0$ would also destroy hyperbolicity.

Define the reduced gravityand use the Boussinesq limit ${\rho }_1\simeq {\rho }_2$ except in buoyancy. The balances becomeThus, for $\mathbf{q}=(g^{\prime },h,u{)}^T$,which is the required entraining shallow-water layer system.