Difficulty understanding Burgers' equation (flow)

Question

Flow of water in a sloping river has velocity $v$. In the simplest case, I know that the resistive force $R=av$ and gravitational force $F=bh$ ($a$ and $b$ are constants). The flow adjusts when the two forces balance, giving $av = bh \implies v = Ch$ ($C$ is a constant).

By mass conservation law, height is determined by $rate(h, x, t)$ where $$h_t + (hv)_x = r.$$ Then I can nicely substitute for $v$ and get $$u_t + uu_x = f,$$ the nonhomogeneous inviscid Burgers’ equation, with $u = 2Ch$ and $f = 2Cr$.

My difficulty is when $R = av^2$ instead. Intuitively, a stronger $R$ means slower flow but how does this affect $u$ and $f$?

Thank you very much!

Answer 1

Assuming that the height of water is positive, $av^2 = bh$ rewrites as $v = c\sqrt{h}$, where $c>0$ is a constant. Thus, $h_t + (hv)_x = r$ yields $$ h_t + c(h^{3/2})_x = r \, , \quad\text{with}\quad h = v^2/c $$ instead of the previous non-homogeneous Burgers' equations $h_t + C(h^2)_x = r$ with $h=v/C$. The new PDE is still a non-homogeneous nonlinear scalar conservation law.