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 $F$.
$$R=av$$ $$F=bh$$
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 first question is how the author got $u$ and $f$ values: how he converted the mass conservation equation to Burger's equation.
If $R=av^2$ instead then $v=\sqrt{\frac{b}{a}h}$ and following the same $v$ substitution, I arrive at
$$h_t +(\sqrt{\frac{b}{a}h}\cdot h)_x = r.$$
Is it legal to simply substitute $h$ with $u$ to get
$$u_t + (\sqrt{\frac{b}{a}}u^{\frac{3}{2}})_x=f\;?$$
Again, I'm having trouble identifying the values for $u$ and $f$. Thank you very much for your time and help!