Chapter 6, 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 $Q=\text{ flow }\left[{\text{m}}^{\text{3}}\text{/s}\right],S=\text{slope [m/m],}$ $H=\text{depth [m], and }n=$ the Manning roughness coefficient. Develop a fixed-point iteration scheme to solve this equation for H 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.