If a water wave with length L moves with velocity across a body of water with depth d, as in the figure, then
(a) If the water is deep, show that v ≈ √gL / (2π).
(b) If the water is shallow, use the Maclaurin series for tanh to show that v ≈ √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 v2 ≈ gd is accurate to within 0.01gL.