Chapter 4, Problem 16P

Manning's formula for a rectangular channel can be written as

$Q=\frac{1}{n}\frac{{\left(BH\right)}^{5/3}}{{\left(B+2H\right)}^{2/3}}\sqrt{S}$

where $Q=\text{ flow }\left(m^3/s\right),n=a$ roughness coefficient, $B=\text{ width }\left(\text{m}\right)\text{, }H=\text{depth }\left(\text{m}\right)\text{,}$ $\text{and }S=\text{slope}$. You are applying this formula to a stream where you know that the $\text{width}=\text{20m}$ and the $\text{depth}=\text{0}\text{.3m}$. Unfortunately, you know the roughness and the slope to only $a\pm 10\%$ precision. That is, you know that the roughness is about 0.03 with a range from 0.027 to 0.033 and the slope is 0.0003 with a range from 0.00027 to 0.00033. Use a first-order error analysis to determine the sensitivity of the flow prediction to each of these two factors. Which one should you attempt to measure with more precision?