How Are Waves Formed?

Tony Butt

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Updated 35d ago

The vast majority of the waves we surf are generated by the wind blowing across the surface of the ocean. In order for those waves to grow, energy must be transferred from the air to the water. Just how this works is not quite as simple as it first sounds, and, in fact, is still not fully understood by scientists. So I’m going to try to explain a couple of theories that were developed several decades ago but are still used in wave-prediction models.

In the very early days, the height of the waves generated by a particular wind blowing across a particular stretch of ocean was calculated empirically. In other words, without the need for much knowledge of any of the physics behind it.

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But then, in the 1950s, scientists John Miles and Owen Phillips started to investigate the physical mechanisms behind ocean wave generation. Together, they came up with a combined theory (called, unsurprisingly, the Miles-Phillips theory) involving two separate mechanisms. The first mechanism generates small waves from a completely flat sea; and the second mechanism – which kicks in later – produces bigger waves on top of the smaller ones. Finally, there is a third stage where limiting factors start to come into play. These inhibit the wave growth so that a natural maximum wave height is reached for a particular windspeed.

In 1957, Owen Phillips developed the first part of the theory – to explain how small waves are generated from a completely flat sea. The key to this theory is that any wind blowing across the surface of the water never blows perfectly horizontally. The air naturally contains random vertical fluctuations. As the wind blows across the water surface, the pattern of random ups and downs changes only very slowly. Therefore, the same pattern travels across the water surface for some distance before morphing into a significantly different pattern, see below.

As a result, the air pushes down on the water in some places and sucks it up in others, which causes the water surface to develop small bumps and dips. These bumps and dips move along as the pattern of ups and downs in the air moves horizontally across the water surface. After a while, the bumps and dips develop into small waves.

Now, these small waves begin to lose their dependence on the air perturbations driving them from above, and start to propagate at their own individual, independent speed. So eventually, most of these waves go out of sync with the air fluctuations and cease to receive energy input from the air. Consequently, the waves generated by this mechanism are only able to reach heights and wavelengths of a few centimetres.

Most of the waves produced by this mechanism are so small that their restoring force – the force that brings them back down after the wind has pushed them up – is the surface tension of the water. For this reason they are called capillary waves. Capillary waves are a precursor to much bigger waves called gravity waves; the ones we actually surf

Capillary waves are a precursor to much bigger waves called gravity waves; the ones we actually surf. They are called gravity waves because (you guessed it) their restoring force is gravity. The mechanism that allows gravity waves to form is different from the first mechanism. It needs a sea surface that already contains small waves before it can get going and, once it does so, the wave growth is exponential.

The second wave-growth mechanism was proposed by John Miles, also in 1957. It extended the work of Phillips to offer a more complete description of how ocean waves are formed.

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The first mechanism was a linear one: the waves grew at a steady rate between zero and a natural limit. But the second mechanism is exponential: the waves grow at a rate which increases with size of the waves themselves. In other words, the bigger they are, the faster they grow. This is because their very presence in the air flow (as they ‘stick up’ more and more out of the water surface) enhances the transfer of energy from air to water. It’s a classic positive feedback loop.

Let’s take a closer look. Think of a wave travelling along with the wind blowing over the top of it. In order for the wave to grow, the air must be travelling faster than the wave. This means that the air exerts a higher pressure at the back of the wave and a lower pressure at the front. It pushes the water surface down at the back and pulls it up at the front. This is what makes the wave grow.

Now as the wave begins to grow and the wave ‘sticks out’ more into the windfield, the pressure increases at the back and decreases at the front. So the wave grows even more. A common way of representing this sort of thing in diagrams is with streamlines, which are a bit like isobars on a weather chart. The streamlines are squashed up at the back of the wave and pulled apart at the front of the wave. As the wave grows they become even more squashed up at the back and pulled apart at the front.

So, as the wave grows, the wind pumps energy into the wave at a faster rate, which makes the wave grow faster. This in turn modifies the windfield even more, which makes the wave grow even faster, and so on. In other words, the wave grows exponentially.

Of course, it would be absurd to think that the waves keep on growing as long as the wind blows. Otherwise you’d get infinitely-high waves wherever there is a constant wind blowing.

There must be a limiting mechanism that stops them growing once they reach a certain height. The maximum height is where the limiting mechanism takes energy out of the waves at the same rate as the wind puts energy in. At this point, the waves will only grow further if the windspeed increases.

Limiting factors, just like the wave-growth mechanisms, are still quite poorly understood. One of the most important is whitecapping. This is where the waves get so steep in the generation area that the top tips over, dissipating a lot of energy in turbulent water motions. Whitecapping starts happening once the wind reaches a particular strength (around force 3 or 4 on the Beaufort scale). The stronger the wind, the bigger the whitecaps.

It follows that for each particular windspeed there is a maximum wave height, reached when the energy taken out by whitecapping equals the energy put in by the wind. The limiting of wave growth by whitecapping is a classic negative feedback mechanism.

More about this in Tony's book Surf Science: an Introduction to Waves for Surfing. Cover shot of Morocco's Anchor Point from Surf Berbere