Surf Science
By Dr. Tony Butt
22nd November 2007


When a swell is first generated in the storm centre, a whole host of different wave heights, periods and directions are produced at the same time, by the wind which is constantly imparting its energy onto the sea surface.  As long as the wind continues to blow over that stretch of the ocean, the waves will continue to be mixed up.  This is a windsea.  
Now, once the waves start leaving the generating area they no longer remain under the influence of the overlying wind, and propagate away as free-travelling swell.  As they travel on their own away from the storm centre, several mechanisms act upon them to change their characteristics.  The two principle things that happen to the waves are (a) they clean up and (b) they get smaller.  As long as they are still in deep water, the waves will get progressively smaller and cleaner the further they travel away from the storm centre.

Swell propagating away from the storm centre depicted as a series of ‘packets’.  As each packet propagates further away from the storm centre, it will spread out in both directions, radially and circumferentially.  The radial spreading causes dispersion of the different wave periods and the circumferential spreading causes the waves to reduce in height.

One of the things that makes water waves so complicated is that they do not all travel at the same speed; they are dispersive. The speed at which a wave travels in deep water is directly proportional to the period of that wave; that is to say, waves with longer periods travel faster.  The speed of an individual wave in deep water in metres per second is equal to 1.56 times the period in seconds, and the speed of a wave group in deep water is 0.78 times the period.  (The swell itself travels at the group speed, not the individual wave speed).  This, the dispersion principle, is probably the most important and fundamental thing to know about the propagation of surface waves across the ocean.  As the swell begins to propagate away from the storm centre, the waves of different periods begin to sort themselves out; the longer, faster ones racing out in front, and the shorter, slower ones lagging behind.  By the time the swell is some distance from the storm centre, the longer ones will have made their way right out in front and the shorter ones will have been left way behind.  The swell has ‘stretched out’, in a radial direction.
Depending on how far the swell has propagated, it can look quite different when it arrives on the coast.  If you were sitting on a coast a short distance from the storm centre (but not inside the storm) you would see the whole swell arriving in a very short time.  Although the long waves would arrive first, the short ones would not be very far behind.  Dispersion would not have had much of a chance to act on the swell, and so the sea might still appear rather messy.  However, if you were, say, 2000 km away from the storm centre, the long waves, being way out in front, would clearly arrive before all the others.  Then all the other waves would turn up, the shortest ones coming in last.  The swell in this case would be cleaner and more lined up, with a smaller number of different wavelengths arriving at any particular time.  Since longer waves travel faster and, hence, carry more energy, the first waves of a new swell are generally the cleanest and most powerful ones, although not necessarily the biggest.
After the swell has peaked in size, the waves do not have the same punch to them as they did before.  At the end of the swell, the last waves to arrive are normally quite slow, close together and not so well lined up.  Sometimes these ‘back-markers’ will fail to arrive altogether, having been attenuated along the propagation path.  Shorter waves are generally more susceptible to being attacked by mechanisms that remove energy, such as friction and opposing winds.

Swells can travel immense distances in the open ocean with very little overall energy loss, particularly when it comes to the long-period waves.  This was shown many years ago with some classic experiments by Walter Munk and colleagues from the Scripps Institute of Oceanography.  In 1957, for example they detected waves at Guadalupe Island off the coast of Baja California waves that had travelled over 15,000 km from a storm in the India Ocean.
Even though there is virtually no loss of energy overall, the waves do get smaller as they propagate away from the storm centre.  This can be understood if we imagine a single wave crest being generated at some point on the ocean, from a wind blowing in a particular direction.  As the wave front moves independently away from the generating position, it will spread out over a progressively wider area.  This process is called circumferential spreading.  It happens because the movement of the water molecules being ‘pushed’ by the wind is transmitted to other molecules to one side of the area being pushed.  But the height of the wave front also gets smaller as it spreads out.  This is because the initial energy given to it when it was generated has to be shared over a progressively wider area.  The only way this can be done is for the wave height to be reduced accordingly.
The width of the wave front, as it spreads out, is directly proportional to the distance it has travelled from the source.  For example, when the swell is 4000 km away from the storm centre it will have spread out over a circumference four times as large as when it was 1000 km away.  If we now think about wave energy instead of height, we can say that the wave energy contained in every metre width of wave front 4000 km away from the storm centre will be four times less concentrated as the wave energy 1000 km away (because it will be four times as spread out).  Now, the relationship between wave energy and wave height is not straightforward; it is quadratic.  If the wave energy is reduced by a factor of four, the wave height will be halved.  So, consider a swell that reaches a point 1000 km away from the storm centre with a height of 3 m.  If the swell continues to propagate and reaches a point 4000 km from the storm centre, it will have spread out four times as far and the energy per metre of wave front will be four times smaller.  If the energy is four times smaller, the height will be halved, namely 1.5 m.

Submitted By Dr. Tony Butt on the 22nd November 2007.
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