River flowing down a slope

Sketch of a gently sloping river with surface height h(x, t)
I am given a simple river system flowing down a slope where the speed of the flow
downstream under the force of gravity builds up until resistive forces $R$ balance the component of gravitational force acting downstream $F$.


The speed will eventually adjust to when $av=bh \implies v=\frac{b}{a}h$. We further assume that rain add water and we let seepage into the ground remove water. This will increase the depth of water at any point in the river at the rate $r(h, x, t)$ and the mass conservation law is given by

$$h_t +(hv)_x = r$$

Using Burger’s equation and substituting for $v$, I get

$$u_t +uu_x =f$$

where $u=2\frac{b}{a}h$ and $f=2\frac{b}{a}r$. My question is how one arrives at the proposed $u$ and $f$ expressions.

If $R=av^2$ instead and following the same procedure I arrived at

$$h_t +(\sqrt{\frac{b}{a}}hh)_x = r$$

Can I reduce this to,

$$u_t +uu_x =f$$

where $u=\frac{3}{2}\sqrt{\frac{b}{a}}h\;\text{and}\;f=\frac{3}{2}\sqrt{\frac{b}{a}}r \;?$

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