?If a water wave with length \(L\) moves with velocity \(v\) across a body of water with

Chapter 11, Problem 35

(choose chapter or problem)

If a water wave with length \(L\) moves with velocity \(v\) across a body of water with depth \(d\), as shown in the figure, then

                                                                   \(v^{2}=\frac{g L}{2 \pi} \tanh \frac{2 \pi d}{L}\)

(a) If the water is deep, show that \(v \approx \sqrt{g L /(2 \pi)}\).

(b) If the water is shallow, use the Maclaurin series for tanh to show that \(v \approx \sqrt{g D}\). (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 \(L>10 d\), then the estimate \(v^{2} \approx g d\) is accurate to within \(0.014 g L\).

                

Equation Transcription:

Text Transcription:

L

v

d

v^2=gL/2pi tanh 2pid/L

v approx sqrt gL/(2pi)

v approx sqrt gD

L>10d

v^2 approx gd

0.014gL