Chapter 8, Problem 20P

The Manning equation can be written for a rectangular open channel as

$Q=\frac{\sqrt{S}{\left(BH\right)}^{5/3}}{n{\left(B+2H\right)}^{2/3}}$

where $Q=$ flow $\left[{\text{m}}^3/\text{s}\right],S=\text{slope }\left[\text{m}/\text{m}\right],H=\text{depth }\left[\text{m}\right],\text{ and }n=$ the Manning roughness coefficient. Develop a fixed-point iteration scheme to solve this equation for $H\text{ given }Q=5,S=0.0002,B=20,\text{ and }n=0.03$. Prove that your scheme converges for all initial guesses greater than or equal to zero.