how to determine the width (x) of a cross section in an elliptical storm drain pipe given a known depth (y)

Question

I am a hydrologist working on a stage-discharge ratings curve for an elliptical storm drain pipe. The pipe is 6.91 ft x 5.35 ft. I have equipment at the site that gives me real-time readings of water depth. To use manning's equation to determine velocity and discharge I need to be able to calculate the width of the cross section given any depth of water in the pipe. So if I know the Y, lets say its 1 ft deep, and I know the dimensions of the pipe stated above, as well as the cross sectional area of any segment of the ellipse given a water depth, how can I find the width of that cross section.

enter image description here

Answer 1

Let $2a=6.91$, $2b=5.35$, so $a$ and $b$ are the semimajor and semiminor axes of the ellipse. The equation of the ellipse is $$ \frac{x^2}{a^2}+\frac{y^2}{b^2} = 1, $$ where the coordinates are from the centre of the ellipse. We have the water depth as $Y=b+y$, since if the water were up to the centre of the ellipse, $Y=b$. The coordinates of the points on ellipse at height $-y$ are $$ \left( a\sqrt{1-\frac{y^2}{b^2}},y \right), $$ and the width of the water is double the $x$ value. Therefore, the width is $$ X=2a\sqrt{1-\frac{(Y-b)^2}{b^2}} = \frac{2a}{b}\sqrt{ 2bY-Y^2 } = \frac{2a}{b}\sqrt{Y(2b-Y)}. $$ (We believe this because it has zeros when $Y=0$ and the pipe is empty, or $Y=2b$ and the pipe is full.) Sticking the numbers in should give you the answer (remember to halve the dimensions as per the first paragraph).