F   Waves II

The physics of deep-water waves

Waves contain energy in two forms: potential energy, and kinetic energy.
The potential energy is the energy required to move all the water from
the troughs to the crests. The kinetic energy is associated with the water
moving around.

People sometimes assume that when the crest of a wave moves across
an ocean at 30miles per hour, the water in that crest must also be moving
at 30miles per hour in the same direction. But this isn’t so. It’s just like
a Mexican wave. When the wave rushes round the stadium, the humans
who are making the wave aren’t themselves moving round the stadium:
they just bob up and down a little. The motion of a piece of water in
the ocean is similar: if you focused on a bit of seaweed floating in the
water as waves go by, you’d see that the seaweed moves up and down,
and also a little to and fro in the direction of travel of the wave – the exact
effect could be recreated in a Mexican wave if people moved like window-
cleaners, polishing a big piece of glass in a circular motion. The wave has
potential energy because of the elevation of the crests above the troughs.
And it has kinetic energy because of the small circular bobbing motion of
the water.

Our rough calculation of the power in ocean waves will require three
ingredients: an estimate of the period T of the waves (the time between
crests), an estimate of the height h of the waves, and a physics formula
that tells us how to work out the speed v of the wave from its period.

The wavelength λ and period of the waves (the distance and time re-
spectively between two adjacent crests) depend on the speed of the wind
that creates the waves, as shown in figure F.1. The height of the waves
doesn’t depend on the windspeed; rather, it depends on how long the
wind has been caressing the water surface.

You can estimate the period of ocean waves by recalling the time be-
tween waves arriving on an ocean beach. Is 10 seconds reasonable? For
the height of ocean waves, let’s assume an amplitude of 1 m, which means
2 m from trough to crest. In waves this high, a man in a dinghy can’t see
beyond the nearest crest when he’s in a trough; I think this height is bigger
than average, but we can revisit this estimate if we decide it’s important.
The speed of deep-water waves is related to the time T between crests by
the physics formula (see Faber (1995), p170):

where g is the acceleration of gravity (9.8 m/s2). For example, if T = 10
seconds, then v = 16 m/s. The wavelength of such a wave – the distance
between crests – is λ = vT = gT2/2π = 160 m.

Figure F.1. Facts about deep-water waves. In all four figures the horizontal axis is the wave speed in m/s. From top to bottom the graphs show: wind speed (in m/s) required to make a wave with this wave speed; period (in seconds) of a wave; wavelength (in m) of a wave; and power density (in kW/m) of a wave with amplitude 1 m.