Physics for Geologists, Second edition

10 Sea waves
The nature of water waves
Waves on water are not the same sort of waves as sound or light, largely
because water is relatively incompressible. There are two main sorts of water
waves; larger waves that are gravitational, and ripples that are due more to
surface tension.
When the wind blows over deep, smooth water, friction tends to push the
surface in the same direction, and irregularities give rise to waves that move
in the same direction as the wind, with their crests normal to it. The first
clear waves are of short wavelength, small height, but relatively steep. The
ratio of wave height to wave length (Hlh), called the steepness, is about
& for these new waves. As the wind continues to blow, the waves become
higher and longer (initially retaining their steepness); but as the wavelength
increases, so does the speed. This acceleration reduces the relative wind
strength (velocity) so the rate of increase of height decreases, and the rate
of increase of wavelength decreases. They become less steep (about &), but
the period (the time between the passage of successive crests through one
position, the inverse of frequency) remains constant.
The maximum height (H,,,) and the maximum wavelength (I,,,) gener-
ated by strong wind blowing in one direction at W knots or m s-I for a couple
of days over a fetch of several hundred kilometres are given approximately
by the following formulae:
H, = w2/165 (m, knots) I,,, = ~ ~ 1 6 (m, . 6 knots)
H, = w2/45 (my m s-*) X, = ~ ~11.74 (m, m s-I).
What is the relationship between wavelength and speed? It could be a func-
tion of viscosity (probably the dimensionless ratio of viscosities of air and
water), g, X, p (the water mass density), and the function must have the
dimensions of velocity, LT-'. Using elementary dimensional analysis,
Copyright 2002 by Richard E. Chapman