Using conservation of mass to derive inviscid Burgers' equation

Question

For a river flowing on a slope, the resistive force is $R = av$ and gravitational force, $F = bh$. The flow speed adjusts itself to $av=bh \implies v=\frac{b}{a}h$. For the mass conservation law,

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

substituting for $v$, one gets $$u_t +uu_x =f \;\text{(inviscid Burgers' equation)}$$ with $u=2\frac{b}{a}h$ and $f=2\frac{b}{a}r$. How is the author getting the second equation (and $u$, $f$ values) from the first equation?

Answer 1

with $$v=\frac{b}{a}h\to v_t=\frac{b}{a}h_t\to h_t=\frac{a}{b}v_t$$ $$(hv)_x=\left(\frac{a}{b}v^2\right)_x=\frac{a}{b}2v v_x$$ thus $$h_t + (hv)_x =\frac{a}{b}v_t+\frac{a}{b}2v v_x=r$$ $$v_t+2v v_x=r\frac{b}{a}$$ with $u=2\dfrac{b}{a}h=2v$ write $$\frac{u_t}{2}+2\frac{u}{2}\frac{u_x}{2}=r\frac{b}{a}$$ so $$u_t +uu_x =f$$ where $f=2\dfrac{b}{a}r$.