= Solution
Here $h$ is <water> depth, $u$ the depth-averaged downstream <speed>, $s$ bed elevation relative to the reference slope, and $\eta=s+h$ free-surface elevation relative to that reference. The constants $\rho_w,\rho_s$ are <water> and grain densities, $S$ is the reference bed slope, $\tau$ bed <shear stress>, $\phi$ bed <porosity>, $c$ <suspended sediment> <mass> <concentration>, $q_b$ bedload solid <volume flux> per unit width, and $v_E,v_D$ erosion and deposition velocities. The settling <speed> is denoted by $v_s$ in the constitutive laws.
Depth <integration> of incompressible <water> conservation gives the depth-storage plus discharge-divergence balance. Depth-averaged <momentum> has local and advective acceleration on the left, and <gravity> driven by the actual surface gradient together with bed drag on the right. <Hydrostatic pressure> and a <shallow-water approximation> underlie this <shallow water equations>[Saint-Venant] description.
<Suspended sediment> conservation first has the conservative form $(hc)_t+(huc)_x=\rho_s(v_E-v_D)$; the <water> balance reduces it to the material-derivative form printed in the source. Solid bed <volume> per unit horizontal <area> is $(1-\phi)s$. Its storage plus bedload divergence equals deposition minus erosion, giving the Exner bed balance. Adding the suspended and bed balances after converting bed <volume> to grain <mass> cancels exchange terms: \b[erosion and deposition transfer sediment between the two reservoirs rather than create it].
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