Why are they so much slower than I thought? Observing the group velocity vs phase velocity of waves

Have you ever seen a speedboat drive past, looked at its wake moving torwards you, then gotten distracted, and when you look back a little while later been surprised that the wake hasn’t moved as far towards you as you thought it would have during the time you looked away?

Well, I definitely have had that happen many times, and the other day I was sitting on the beach with a friend and we talked about why you initially perceive the waves moving a lot faster than they turn out to be moving in the end. While I didn’t film it then, I’ve been putting my time on the GEOF105 student cruise to good use to check out waves in addition to the cool research going on on the cruise, so now I have a movie showing a similar situation!

But let’s talk a little theory first.

Phase velocity

The phase velocity of a wave is the speed with which you see a wave crest moving.

Waves can be classified into long vs short waves, or deep- vs shallow water waves. But neither deep and shallow, nor long and short are absolute values: They refer to how long a wave is relative to the depth of the water in which it is moving. For short or deep water waves, the wavelength is short relative to the water depth (but can still be tens or even hundreds of meters long if the water is sufficiently deep!). For long or shallow water waves, the wave length is long compared to the water depth (for example Tsunamis are shallow water waves, even though the ocean is on average about 4 km deep).

For those long waves, or shallow water waves, the phase velocity is a function of the water depth, meaning that all shallow water waves all move at the same velocity.

However, what you typically see are deep water waves, and here things are a little more complicated. Since phase velocity depends on wave length, it is different for different waves. That means that there is interference between waves, even when they are travelling in the same direction. So what you end up seeing is the result of many different waves all mixed together.

If you watch the gif below (and if it isn’t moving just give it a little moment to fully load, it should then start), do you see how waves seem to be moving quite fast past the RV Harald Brattstrøm, but once you focus on individual wave crests, they seem to get lost, and the whole field moves more slowly than you initially thought?

That’s the effect caused by the interference of all those waves with slightly different wave lengths, and it’s called the group velocity.

Group velocity

The group velocity is the slower velocity with which you see a wave field propagate. It’s 1/2 of the phase velocity, and it is the velocity with which the signal of a wave field actually propagates. So even though you initially observed wave crests moving across the gif above fairly quickly, the signal of “wave field coming through!” only propagates with half the phase velocity.

Usually you learn about phase and group velocities in a theoretical way and are maybe shown some animations, but I thought it was pretty cool to be able to observe it “in situ!” :-)

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