Example: Consider a small solitary wave that moves from right to left on the

surface of a water channel of undisturbed depth h$.$ By assuming a small change

in the surface elevation across the wave, derive an expression for its propagation

speed, U$,$ when the channel bed is flat and frictionless.

The Answer is U$^2=($g$/2)($h out$/$h in$)($h in$+$h out$)$

you can assume the water is l into the page, $($h in$)$ is on the left, and $($h out$)$ is on the right.

Please show steps, and possibly explanations.