# Phase and group velocities in deep and shallow water

When Tor came to visit me in GFI’s basement lab a couple of days ago, he told me about an experiment he had seen in Gothenburg in the seventies. So Elin and I obviously had to recreate it on the spot. Therefore today, we are comparing phase- and group velocities in deep and shallow water!

Waves are excited by means of an oscillating, hand-helt beer can, curtesy of the beer brewing club at GFI. The experiments are filmed and wave lengths and phase velocities are determined from the videos, which is a lot easier than measuring them directly while the experiment is being run.

# Shallow water waves

For shallow water, we are using a water depth of 10 cm. Waves are very easy to see and phase velocities are equally easy to measure.

There is another experiment on (standing) shallow water waves being run at GFI the year before students attend GEOF213, which I described back in 2013.

# Deep water waves

For deep water waves, we use a water depth of 42.5 cm (the exact number only matters when the tank filling is also used to fiddle with the dead water experiment, as I had been when the idea for this experiment came up).

Typical wave lengths that are easy to do are between 10 and 25 cm (wave lengths obviously have to be short enough that the water is still “deep”, i.e. H>>wave length) — Elin’s instruction to me for the kind of waves she wanted was “Allegro!” :-D Elin, you are really the coolest and most fun person to play with tanks with!

In deep water, we now have the added difficulty that the phase speed is twice as fast as the group speed. This makes observing the whole thing a lot more difficult. Also amplitudes are a lot smaller now, since the tank was so full and we wanted to keep the water inside…

Here is t0 — Elin has just dipped the beer can into the water for the first time

t1 — can you see the wave signal has propagated up to where the red arrow is pointing to?

t2 — the signal has reached my thumb at the left edge of the picture.

From timing this, we can calculate the group speed. We can also measure the wave length on the video and then calculate a theoretical phase speed from that. For the experiments Elin and I did, the results were pretty good, as in phase speed was usually about twice as fast as group speed. And I am curious to hear how well this works out when the students run the experiment!

# Playing with roll wave animations

I love how powerful Powerpoint is, at the same time there surely is a way out there to create these kind of animations with a little less copy & pasting, and especially without manually moving tons of stuff by juuust a tiny little bit from frame to frame?

How would you build these kinds of pictures? I’m even considering Matlab at this point (which I really don’t think would be such a stupid idea after all)

This is an animated gif. If it isn’t playing, I have no idea why not… It is playing on Twitter (link here)

# I’m explaining your wave pictures: A #friendlywave from the Île d’Yeu, France

The whole #friendlywave thing (where I explain your wave picture) is starting off great! Here is one that reached my via my Twitter; link to thread here.

What’s going on in the north east of Île d’Yeu, France? Here are four pictures from the Twitter thread that got me intrigued: Because of the awesome waves they were displaying, but also because they introduced me to ESA’s EO browser which is so amazing that I don’t think I will be able to stop playing anytime soon!

First, a true color image of the Île d’Yeu and, more importantly, the wave field around it (Click on all pictures to enlarge).

And this is what the topography in that area looks like:

Zooming in on the area north of the eastern tip where something interesting is happening……this checkerboard pattern of waves! Now the question is what causes those waves. Well, let’s find out, shall we?

I couldn’t figure out exactly where the image above was from, but I am seeing a very similar pattern in the pictures that I saved off the EO browser myself.

First, here is a true color image again (click to enlarge, or click the link to see it on the browser to play yourself)

True color image of Île d’Yeu and surrounding ocean, acquired with EO browser, January 28th, 2019.

Here is the same image, except with my annotations on it. I have marked a couple of wave crests to show what I think is going on. What I see here (and please let’s discuss this! I’m super curious to hear what you think!) is a wave field coming in from west northwest-ish (see straight-ish fronts on the top left). When this wave field encounters an obstacle in its path (the island), it gets diffracted, kind of as if there were two very wide slits on either side of the island (a very simple example of that here). It’s difficult to follow the wave crests that pass the island on its north side, but the ones that go round the south side are clearly visible as they turn around the eastern tip of the island.

Zooming in to look at it more closely:

True color image of Île d’Yeu and surrounding ocean, acquired with EO browser, January 28th, 2019.

And here is my annotated version of the wave field. You recognise the wave crests that were propagating along the southern side of the island, then turned around the eastern tip and are now spreading northward. And you see the wave crests of the waves that travelled along the north coast all along. Notice how they are crossing in a crisscross pattern?The area with the really dense red checkered pattern is the one I think was shown on the original picture on Twitter. So my interpretation is that it’s an interference pattern of waves, all originating in the same wave field, being diffracted l’Île de Yeu. What do you think? Do you agree?

What I find quite interesting is that it’s very easy to follow the crests that propagate northward around the eastern tip, but a lot more difficult to do the same for the ones propagating southward. I could imagine that the explanation is the topography: The waves propagating in the north of the island were in shallower water for pretty much the length of the island, so they might have lost a lot of their energy already, whereas the ones from the south only run into shallower water once they’ve turned around the eastern tip of the island.

Thanks, Rémi, for pointing me to ESA’s awesome EO browser and to your super interesting Twitter!

P.S.: Speaking of topography: Of course the change in water depth could also have an effect on the wave field by refracting the waves towards the slower medium, i.e. the shallower water. But I don’t think that’s the case here. Do you?

# Roll waves, one of the more complicated #friendlywaves I’ve gotten over the years…

#friendlywave is the new hashtag I am currently establishing. Send me your picture of waves, I will do my best to explain what’s going on there!

When it rains, it pours, especially in LA. So much so that they have flood control channels running throughout the city even though they are only needed a couple of days every year. But when they are needed, they should be a tourist attraction because of the awesome wave watching to be done there! As you see below, there are waves — with fronts perpendicular to the direction of flow and a jump in surface height — coming down the channel at pretty regular intervals.

Roll waves on Verdugo Wash. Photo by Mike Malaska

Even though this looks very familiar from how rain flows in gutters or even down window panes, having this #friendlywave sent to me was the first time I actually looked into these kinds of flows. Because what’s happening here is nothing like what happens in the open ocean, so many of the theories I am used to don’t actually apply here.

# Looks like tidal bores traveling up a river

The waves in the picture above almost look like the tidal bores one might now from rivers like the Severn in the UK (I really want to go there bore watching some day!). Except that bores travel upstream and thus against the current, and in the picture both the flow and the waves are coming at us. But let’s look at tidal bores for a minute first anyway, because they are a good way to get into some of the concepts we’ll need later to understand roll waves, like for example the Froude number.

## Froude number: Who’s faster, current or waves?

If you have a wave running up a river (as in: running against the current), there are several different scenarios, and the “Froude number” is often used to characterize them. The Froude number Fr=u/c compares how fast a current is flowing (u) with how fast a wave can propagate (c).

### Side note: How do we know how fast the waves should be propagating?

The “c” that is usually used in calculating the Froude number is the phase velocity of shallow water waves c=sqrt(gH), which only depends on water depth H (and, as Mike would point out, on the gravitational constant g, which I don’t actually see as variable since I am used to working on Earth). (There is, btw, a fun experiment we did with students to learn about the phase speed of shallow water waves.) This is, however, a problem in our case since we are operating in very shallow water and the equation above assumes a sinusoidal surface, small amplitude and a lot of other stuff that is clearly not given in the see-saw waves we observe. And then this stuff quickly gets very non-linear… So using this Froude number definition is … questionable. Therefore the literature I’ve seen on the topic sometimes uses a different dispersion relation. But I like this one because it’s easy and works kinda well enough for my purposes (which is just to get a general idea of what’s going on).

Back to the Froude number.

If Fr<1 it means that the waves propagate faster than the river is flowing, so if you are standing next to the river wave watching, you will see the waves propagating upstream.

Find that hard to imagine? Imagine you are walking on an elevator, the wrong way round. The elevator is moving downward, you are trying to get upstairs anyway. But if you run faster than the elevator, you will eventually get up that way, too! This is what that looks like:

If Fr>1 however, the river is flowing faster than waves can propagate, so even though the waves are technically moving upstream when the water is used as a reference, an observer will see them moving downstream, albeit more slowly than the water itself, or a stick one might have thrown in.

On an escalator, this is what Fr>1 looks like:

But then there is a special case, in which Fr=1.

## Hydraulic jumps

Fr=1 means that the current and the waves are moving at exactly the same velocity, so a wave is trapped in place. We see that a lot on weirs, for example, and there are plenty of posts on this blog where I’ve shown different examples of the so-called hydraulic jumps.

See? In all these pictures above there is one spot where the current is exactly as fast as the waves propagating against it, and in that spot the flow regime changes dramatically, and there is literally a jump in surface height, for example from shooting away from where the jet from the hose hits the bottom of the tank to flowing more slowly and in a thicker layer further out. However, all these hydraulic jumps stay in pretty much the same position over pretty long times. This is not what we observe with tidal bores.

On an escalator, you would be walking up and up and up, yet staying in place. Like so:

## Roll waves

Tidal bores, and the hydraulic jumps associated with their leading edges, propagate upstream. But they are not waves the way we usually think about waves with particles moving in elliptical orbits. Instead, they are waves that are constantly breaking. And this is how they are able to move upstream: At their base, the wave is moving as fast as the river is flowing, i.e. Fr=1, so the base would stay put. As the base is constantly being pushed back downstream while running upstream at full force, the top of the wave is trying to move forward, too, moving over the base into the space where there is no base underneath it any more, hence collapsing forward. The top of the wave is able to move faster because it’s in “deeper” water and c is a function of depth. This is the breaking, the rolling of those waves. The front rolls up the rivers, entraining a lot of air, causing a lot of turbulent mixing as it is moving forward. And all in all, the whole thing looks fairly similar to what we saw in the picture above from Verdugo Wash.

# But the waves are actually traveling DOWN the river

However there is a small issue that’s different. While tidal bores travel UP a river, the roll waves on Verdugo Wash actually travel DOWN. If the current and the waves are traveling in the same direction, what makes the waves break instead of just ride along on the current?

## What’s tripping up these roll waves?

Any literature on the topic says that roll waves can occur for Fr>2, so any current that is twice as fast as the speed of waves at that water depth, or faster, will have those periodic surges coming downstream. But why? It doesn’t have the current pulling the base away from underneath it as it has in case of a wave traveling against the current, so what’s going on here? One thing is that roll waves occur on a slope rather than on a more or less level surface. Therefore the Froude number definitions for roll waves include the steepness of the slope — the steeper, the easier it is to trip up the waves.

### Shock waves: Faster than the speed of sound

Usually shock waves are defined as disturbances that move faster than the local speed of sound in a medium, which means that it moves faster than information about its impending arrival can travel and thus there isn’t any interaction with a shock wave until it’s there and things change dramatically. This definition also works for waves traveling on the free surface of the water (rather than as a pressure wave inside the water), and describe what we see with those roll waves. Everything looks like business as usual until all of a sudden there is a jump in the surface elevation and a different flow regime surging past.

If you look at such a current (for example in the video below), you can clearly see that there are two different types of waves: The ones that behave the way you would expect (propagating with their normal wave speed [i.e. the “speed of sound”, c] while being washed downstream by the current) and then roll waves [i.e. “shock waves” with a breaking, rolling front] that surge down much faster and swallow up all the small waves in their large jump in surface elevation.

Video by Mike Malaska

In the escalator example, it would look something like this: People walking down with speed c, then someone tumbling down with speed 2c, collecting more and more people as he tumbles past. People upstream of the tumbling move more slowly (better be safe than sorry? No happy blue people were hurt in the production of the video below!).

Looking at that escalator clip, it’s also easy to imagine that wave lengths of roll waves become longer and longer the further downstream you go, because as they bump into “ordinary” waves when they are about to swallow them, they push them forward, thus extending their crest just a little more forward. And as the jump in surface height gets more pronounced over time and they collect more and more water in their crests, the bottom drag is losing more and more of its importance. Which means that the roll waves get faster and faster, the further they propagate downstream.

Speaking of bottom drag: When calculating the speed of roll waves, another variable that needs to be considered is the roughness of the ground. It’s easy to see that that would have an influence on shallow water. Explaining that is beyond this blog post, but there are examples in the videos Mike sent me, so I’ll write a blogpost on that soon.

So. This is what’s going on in LA when it is raining. Make sense so far? Great! Then we can move on to more posts on a couple of details that Mike noticed when observing the roll waves, like for example what happens to roll waves when two overflow channels run into each other and combine, or what happens when they hit an obstacle and get reflected.

Thanks for sharing your observations and getting me hooked on exploring this cool phenomenon, Mike!

# 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!” :-)

# Wave watching: Refraction and diffraction of waves

A little more wave watching, today with a focus on how waves change direction when they run into shallow water. Let’s look at this beautiful wave and see what happens when it reaches the shallow shore.

Above, you see the wake of the pilot ship, consisting of many wavelets that propagate as parallel wave crests towards the shore.

Below, you see that the wave is propagating at an angle to the shore (something around 45 degrees, maybe?). If you focus on the wave crest that is just offshore of that little obstacle in the water (curious enough, a piece of brick wall), you clearly observe that angle. But then looking at the next wave crest in-shore, it is almost parallel to the shore! Assuming that both crests come from the same wave field, so that the second one was in the same position as the other one only moments before (which I know it was because I observed it), something clearly happened between then and now.

# Refraction of waves

Why do waves change direction as the water depth changes? As waves run from deep into shallow water, at some point they start to “feel” the bottom, which slows them down.

Or, more scientifically speaking, the dispersion relation for shallow water waves is a function of water depth: The shallower the water, the slower the waves. That means that if a wave crest is running on a slope with one side being in shallower water while the other one is still in deeper water, it will change direction towards the shallow water because the shallow side of the crest is slowed down while the deeper side keeps on moving faster, thus forcing the whole crest around a curve.

But in this picture series there is more to see: See how the wave crest gets deformed after it has passed that obstacle?

# Diffraction of waves

This is a process called diffraction: The change of direction after a wave crest has passed either through a slit and then starts radiating from that slit as circle segments, or, in this case, an obstacle. The wave passing an obstacle is, in a way, the same as the wave passing through two wide slits which are very close to each other, only separated by the obstacle: The edges of the wave crest at the edges of the “slits” also start radiating out as circle segments!

One spot, so many things to observe!

And there are, of course, ships. What I wanted to show on this picture is a close-up of the turbulent wake of the ship, but it’s really difficult to see so I’ll let that pass for today.

And the picture below shows so much cool stuff: Waves radiating from that pylon. Ripples on the surface by a gust of wind. Wave crests getting a lot steeper as they run up on the slope. And, my main reason for posting: I really like how the wave is spilling as it breaks! :-)

# 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

# Watch the dispersion relation in action

Remember how we talked about how waves seem to propagate extremely slowly into that calm patch that occurs when a boat pulls away from a dock? Well, the other day I noticed that there is even more physics you can see when watching a similar situation: You see how long waves propagate much faster than short waves (that is for deep water waves, in shallow water the wave speed only depends on water depth, not on wave length)

Here you see a formerly smooth patch of water where the Håkon Mosby was until a minute ago, and you see how long waves have already propagated into that smooth patch while shorter waves are everywhere in the choppy water around the smooth patch, but have yet to propagate into it. Now that I think of it I’ve seen this many times before, I just never noticed. It’s even visible in the video I posted with the other blog post.

And here is a video. Note how the long waves invade the smooth spot of water long before the shorter waves do: