# Why and how are wave sets (groups) created?

At a coasts wave normally arrive in groups (sets) and we wondered why this is the case. Some internet research did reveal some clues but no convincing answer.

I want to know why waves gather in groups (sets) and what physical background is responsible for this.

I list some «facts» I found as a starting point:

It's said that:

• sets occur when different swells (originating from different storms) travel into the same direction and merge together: if a wave's crest coincides with another crest a bigger wave results, if the wave's peak hits a trough then a cancelling effect will make the sea go flat. Source: The Stormrider Guide: Europe The Continent. -> Does that mean if we have a single source of wave energy (1 swell) we will never see any sets? And does this also mean that in between sets we see the result of two waves having eleminated themselves?
• The same book also states that faster waves and slower waves group somehow. Faster waves with longer wavelengths overtake smaller waves and the longer the waves travel the more distinct is the grouping. But why do they travel in groups. It makes sense to me that the faster go up front but why do they organize? I read something about the groupiness factor but I don't really understand what this means. And all this still makes no set, right? It just explains, why we can measure distinct waves after the wave has travelled for some time?
• I also read that one can imagine that a set is also kind of a part of a bigger wave. The first set wave being kind of small, increasing in height to the wave in the middle and then decreasing again until the last and smallest wave of the set comes in. Is this a reason why the set seems to be hold together? And if yes, why?
• In this article, they say that the last waves in a set travel faster than the ones in front and that they increase in height until the get slower again (and are in front now). Does anyone has an explanation for this?

It's written

Although individual waves move at twice the speed of the group, they are bound to it by the energy they all share.

How can this be explained. Why can't a wave break out of a group and travelling a bit faster.

• And in general why are there groups in the first place. As written in the same article:

Wave measurements usually show a tendency for large waves to group together--often referred to by scientists as "groupiness."

Does that mean we can see the groups already or is it just group after group without any space between? Because if the pattern looks like this:

|group| ------- |group| ------- |group|

instead of

|group||group||group||group|

Where does the space in between result from? Also, if the first pattern is correct, this would directly contradict the statement from before, that sets result from two or more interfering swells...

I would be glad if somebody could explain this a bit more into detail, with keeping it understandable. :)

Thanks in advance.

Cheers

## 1 Answer

To respond to the part:

Although individual waves move at twice the speed of the group, they are bound to it by the energy they all share.

-> all the waves can be subjected to the phenomenon called dispersion (depends on the problem and prerequisites for the dispersion's development, but in your situation gravity plays a role for the frequency dispersion of the waves). What you describe here falls under the physics of dispersion: you have a wave travelling packet propagating with a certain speed (called the group speed) and individual waves propagating with the phase speed. In classical (normal) dispersion case the group speed is slower than the phase speed of each individual wave. The phase speed of each wave depends on the wavelength of this wave, and in case of water waves the water depth. You have different formulas for such dependenicis in deep or shallow waters. For a fixed water depth, waves with a larger wavelength will propagate faster than shorter waves, which again will come out of this.

In a packet, group velocity and energy velocity are closely related, while in the lossless media they are actually equal. Hence, the energy velocity is not defined by the phase velocity of each individual wave in your packet, but by the propagation speed of the packet as a whole. Hence, when taking about the energy transport, you consider a packet and not each individual wave.

And finally, for water waves you also have amplitude dispersion, where waves of larger amplitudes have a different phase speed from small-amplitude waves. I don't know about oceanography for sure, but commonly it is related to the phenomenon of attenuation in lossy media.

In the end to describe such a waves' behavior, typically superposition of sinusoidal waves with different wavelengths, initial phases and amplitudes is considered, while some boundary conditions are taken into account. Dispersion relations can then be extracted for each situation (e.g. shallow/deep water etc). So each wave that you observe in nature is a complex superposition of many sinusoidal waves.

In order to find answers to your questions, start with investigating the physics of dispersion of water waves.