# 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!” :-)

# Shallow water waves

Have you ever noticed how, if you are at a shallow beach, no matter how choppy waves are further offshore, everything becomes nice and orderly on the beach?

Below you see where the water depth suddenly increases, both from the color of the water and from the wave pattern. While in deeper water waves propagate at all kinds of speeds depending on their wavelength, the moment the water becomes shallow enough, all waves propagate at the same speed (except for the really short waves for which the water is still deep, but let’s forget about those). If all waves propagate at the same speed, it means that the form of the wave that we observe stays constant over time and just moves as a whole. Hence it looks a lot more tidy than the choppy waves further out.

Funny that in all these years of wave watching, I have never thought about that before!

Watch the movie below to see for yourself

# Group velocity and phase velocity

When I recently wrote about observing waves in a different way, I talked about light being focussed by the waves on the sea floor.

In the other post, I focussed on how looking at the light and dark pattern on the sea floor makes waves visible that are otherwise hard to see when just looking at the surface of the water:

But it also makes something else easier to notice, and that is how phase velocity and group velocity are really different. We know they should be, but on a choppy water surface it is really difficult to keep track of individual waves as they wobble through each other. But looking at the light and dark pattern on the sea floor, it actually becomes easier to observe. See those brighter areas and darker areas? Wave groups!

And in the movie below, you’ll see how they eat up waves that run into them, and how other wave crests come out in the front as they are overtaking the group. Cool! :-)

# Why waves propagate so slowly into smooth patches of water

The morning I went to Heligoland I spent some time in the port of Hamburg, trying to film a phenomenon I had recently chatted about with the author of this inspiring guest post: How waves seem to propagate super slowly into smooth patches of water. It turned out to be really difficult to film (because ships didn’t go where I expected them to go [you see me walk a couple of steps half way through the video below, because I needed to get away where a boat was docking], other ships cross the water you are filming, and because filming water is pretty difficult in general).

Here is my best attempt:

So why does it seem to take waves so surprisingly long to propagate into smooth patches of water? Well, because what we see and notice is the phase speed of waves, with which the crests propagate. But the wave field itself only propagates with group speed, which is half the phase speed. So from the movement we notice, waves should be invading the smooth patches twice as fast as they actually do!

Now I need to go and find a good way to film this phenomenon…

# Signal velocity

How can a signal travel faster than the phase of a wave, or individual particles?

I remember having a really hard time with the concept of a signal traveling faster than the phase of a wave or than individual particles when I first heard about it during my first year at university. I know my physics professor had an example he thought would help us, and I remember that it was something about being on a playground and stepping on something, but I remember that even then I didn’t get what point he was trying to make.

Anyway. I have blinds in my living room, and whenever I open or close them, I somehow think about this. In the movie below you’ll see me crank the blinds up and down. From the reflection of the lit door in the background you can see that the camera stays in more or less the same position during the movie (yes, dad, next time I’ll use a tripod!), and from the sound you hear that I’m cranking with more or less the same rate throughout the movie. And yet you see the blinds seemingly move with two different velocities: One when all the panels move in parallel, and one when the signal that something started moving (or stopped, as in the second case) propagates through the blinds as the gaps between the panels open or close.

Now tell me: Is this a good example? Or why not? What would be better?