## The speed of water waves

Teaching Guidance for 11-14 14-16

In water whose depth is large compared to the wavelength, the wave speed expression contains two terms, one for gravity effects and one for surface tension effects. The wave speed is given by:

*v*^{ 2} = g*λ*2π+2πγ*λ**ρ*

where *g* is the gravitational field strength, γ is the surface tension, *ρ* is the density of the water, and *λ* the wavelength. As this equation makes clear (wave speed depends on wavelength), water is a dispersive medium.

For short wavelength (ripples), the second term predominates, and the speed is approximately

*v* = *2πγ* *λ**ρ*

Tiny ripples on water have a speed which depends on the wavelength, *λ*. Surface tension forces move these waves along. There are gravitational forces on the tiny humps of water, but they are too small to matter.

For long waves, in deep water, the first term predominates, and the speed is approximately

*v* = *g λ*

*2π*

In deep water, the surface tension, γ, is too small to matter. The density, *ρ*, does not appear because if it increases, the force acting and the mass to be moved both increase by the same factor, with no net effect on the response time of water ahead of a wave front.

The speed of waves in shallow water can be given by *v*_{shallow}≈√ *gh* (assuming *λ* >> *h* and *A* << *h*, where *A* is the wave amplitude, and *h* is simply the depth of the water).

See Tricker, R. A. R. (1964), *Bores, Breakers, Waves and Wakes* or Barber, N. F. & Whey, G. (1969), *Water Waves*.

Bores are a special case of shallow water waves . A bore can easily be made in a long narrow water trough by sweeping water along at a steady rate using a wide paddle.