Chapter 11.11, Problem 35E

If a water wave with length L moves with velocity v across a body of water with depth d, as in the figure on page 822, then

$v^2=\frac{gL}{2\pi }\tanh \frac{2\pi d}{L}$

(a) If the water is deep, show that $v\approx \sqrt{gL/\left(2\pi \right)}$.

(b) If the water is shallow, use the Maclaurin series for $\tanh$ to show that $v\approx \sqrt{gd}$. (Thus in shallow water the velocity of a wave tends to be independent of the length of the wave.)

(c) Use the Alternating Series Estimation Theorem to show that if $L>10d$, then the estimate $v^2\approx gd$ is accurate to within $0.014gL$.