Why is the speed of water waves dependent on depth?

[quantum123]

Why is the speed of water waves dependent on the depth?

 

[Studiot]

The velocity, c, of a simple sinusoidal surface water wave is described by, where L is the wavelength and d the water depth:

$$c = \sqrt {\frac{{gL\tanh \left( {\frac{{2\pi d}}{L}} \right)}}{{2\pi }}}$$

Note that when d > L /2

$$\tanh \left( {\frac{{2\pi d}}{L}} \right) \approx 1$$

so the velocity for deep water reduces to

$$c = \sqrt {\frac{{gL}}{{2\pi }}}$$

and when d << L/2

$$\tanh \left( {\frac{{2\pi d}}{L}} \right) \approx \left( {\frac{{2\pi d}}{L}} \right)$$

So the velocity becomes

$$c = \sqrt {gd}$$

This is because the water particles are moving in (nearly) circular orbits in deep water. As the water shoals the bottom exerts a drag which elongates the orbit to an ellipse, which gets flatter and flatter with shoaling.

 

[HallsofIvy]

Roughly speaking, friction with the bottom slows the wave so that waves in deeper water are faster. The more precise derivation, showing that the wave speed is proportional to the square root of the depth is what studiot gives.