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Numerade Educator

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Problem 49 Medium Difficulty

If a water wave with length $ L $ moves with velocity $ v $ in a body of water with depth $ d, $ then
$ v = \sqrt {\frac {gL}{2 \pi} \tanh (\frac {2 \pi d}{L}} $
where $ g $ is the acceleration due to gravity. (See Figure 5.) Explain why the approximation
$ v \approx \sqrt {\frac {gL}{2 \pi}} $ is appropriate in deep water.

Answer

$\sqrt{\frac{g L}{2 \pi}}$

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Video Transcript

Okay, we know that tan of two pi d over l We know we can write this as the limit as D approaches positive infinity of each of the four Pi di over Al to the minus one. This simplifies to the square root of G l over to pie. Now remember, that is the duct increases the function 10 h of two pied de over Al goes towards one though for the velocity as she out over too pie.