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!