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In the characteristic coordinates, the chain rule gives ${\partial }_x={\partial }_{\xi }+{\partial }_{\eta }$ and ${\partial }_t=c({\partial }_{\xi }-{\partial }_{\eta })$. HenceIntegrating this mixed derivative gives $\zeta =F(\xi )+G(\eta )$, a sum of oppositely traveling profiles. The initial data require $F+G=u_0$ and $c(F^{\prime }-G^{\prime })=v_0$, soIntegrating the profiles and matching their total additive constant gives the D'Alembert formula with initial velocity:Here $u_0$ names the initial surface displacement, rather than the water velocity $u$ appearing in the original shallow water equations. The formula verifies both initial conditions directly.

The Boussinesq approximation requires $\delta =(\rho -{\rho }_0)/{\rho }_0=\varphi ({\rho }_p/{\rho }_0-1)\ll 1$. The reference mass density ${\rho }_0$ can then be used in inertia, while the small density excess is retained in buoyancy. Dilute particle volume fraction $\varphi \ll 1$ alone does not suffice if ${\rho }_p/{\rho }_0$ is exceptionally large. Define $G=g({\rho }_p-{\rho }_0)/{\rho }_0$, so the reduced gravity is $g^{\prime }=G\varphi$.

The shallow water equations require depth small compared with the horizontal evolution scale, weak vertical acceleration and approximately hydrostatic pressure. Here the ambient is deep and quiescent, so its leading contribution is a hydrostatic reference pressure; the excess pressure in the layer is ${\rho }_{0}G\varphi (h-z)$. The model also needs nearly uniform streamwise velocity across the section and the stated absence of secondary circulation. The advancing gravity current nose is a separate region where these assumptions can fail.

A nearly uniform particle volume fraction can be maintained by turbulent mixing with eddy diffusivity $K_z$, provided $h^2/K_z\ll h/w$ and the mixing time is short compared with horizontal advection and bulk evolution. Equivalently, $wh/K_z\ll 1$ keeps the bulk settling-induced gradient small. The particles should respond rapidly enough to follow the mixing motions. A thin boundary layer adjacent to the absorbing bed or walls need not be well mixed.

Uniform bulk concentration does not imply zero sedimentation. The relative downward particle velocity still supplies a particle deposition flux of approximately $w\varphi$ to an absorbing boundary. Mixing redistributes the remaining particles and maintains the bulk concentration profile while its overall level decreases; it need not suspend every particle indefinitely. Deposition requires negligible resuspension in the model.

Linearizing the shallow water equations about depth $h$ at rest gives wave speed $c=\sqrt{gh}$. Replacing $h$ by $c$ in the nonlinear equations yields $c_t+u{c}_x+(c/2)u_x=0$ and $u_t+u{u}_x+2c{c}_x=0$. Direct addition and subtraction giveThus the Riemann invariants $J_{\pm }=u\pm 2(c-c_0)$ are constant on the characteristics $C_{\pm }$ with speeds $u\pm c$. In this problem $c_0=\sqrt{g{h}_0}$.