Derive kinematic wave approximation

Starting point

The momentum equation of the Saint-Venant system is

$$\underbrace{\frac{\partial Q}{\partial t}}_{1}+\underbrace{\frac{\partial}{\partial x}\left(\frac{Q^2}{A}\right)}_{2}+\underbrace{gA\frac{\partial y}{\partial x}}_{3}-gA(S_0-S_f)=0$$

Dividing by $gA$ gives the form with slopes: $S_f=S_0-\dfrac{\partial y}{\partial x}-\dfrac{V}{g}\dfrac{\partial V}{\partial x}-\dfrac1g\dfrac{\partial V}{\partial t}$, in which terms 1, 2 and 3 are the local acceleration, convective acceleration and pressure terms.

Approximation

For flood waves on steep channels these three terms are very small compared with $S_0$ and $S_f$. Neglecting them:

$$S_0-S_f=0\ \Rightarrow\ S_f=S_0$$

The flow is then locally uniform: the discharge depends only on the area at the same section. Using Manning's equation $Q=\dfrac{1}{n}\dfrac{A^{5/3}}{P^{2/3}}S_0^{1/2}$,

$$Q=\alpha_kA^{\beta},\qquad \beta=\frac53,\ \ \alpha_k=\frac{S_0^{1/2}}{nP^{2/3}}$$

Kinematic wave equation

Continuity: $\dfrac{\partial A}{\partial t}+\dfrac{\partial Q}{\partial x}=q$. From the relation above, $\dfrac{\partial Q}{\partial x}=\dfrac{dQ}{dA}\dfrac{\partial A}{\partial x}$, hence

$$\frac{\partial A}{\partial t}+c\frac{\partial A}{\partial x}=q,\qquad\text{or}\qquad \frac{\partial Q}{\partial t}+c\frac{\partial Q}{\partial x}=c\,q,\qquad c=\frac{dQ}{dA}=\alpha_k\beta A^{\beta-1}=\beta V$$

A flood wave therefore moves downstream at the kinematic celerity $c\approx\frac53V$ without changing shape (no attenuation or backwater effects).