# One of the most difficult #friendlywaves I’ve ever gotten! Did I get it right?

Florian sent me a #friendlywave — a wave picture he took, with hopes that I might be able to explain what is going on there. And this one had me puzzled for some time!

This is what the picture looks like:

What I knew about it: Florian was on the ferry from Wisschafen to Glückstadt, crossing the Elbe river.

In the picture itself, there are several features that jumped at me. First, drawn in with the lightblue line below: A sand bank parallel(-ish) to the island’s coast line.

Then, the ship’s wake (shown in red) breaking right near the ship (orange) and turning (green) and breaking (yellow) where it runs on the sand bank.

Florian wrote he was watching the ferry’s wake and noticed something curious: There seemed to be a shallow part, where the waves suddenly became a lot faster! And could I explain what was going on?

Looking at the picture, there were two possibilities for what he might have meant (and, spoiler alert — I completely jumped on the wrong one first!).

Below, I’ve drawn in the part of the wake that is running on the shallow sand bank (green) and how those wave crests continue on the other side of the sand bank (red). I’ve also drawn in some mystery wave crests in blue. Those were the ones I chose to focus on first, since Florian had written that he noticed waves behaving weirdly and suddenly becoming much faster. So if we are talking fast, we are talking really fast, right?

So how do we explain those blue wave crests?

There is a limit for the maximum speed a wave can have. That limit depends on the wave’s wave length: The longer a wave, the faster it travels. In deep water, i.e. water deeper than 1/2 the wavelength, the wave travels at this maximum speed (see green lines in the plot below).

But as it comes into shallower water, it gets slowed down (see black lines in the plot below — those are just a quick sketch, there are complicated equations to calculate it exactly).

In shallow water, i.e. water that is smaller than 1/20th of the wave length, the phase speed only depends on water depth: The shallower the water, the more the wave is being slowed down (see the red lines in the plot below).

Sorry about the quality of the sketch — I don’t have Matlab or anything else useful on the computer I have available right now, so I drew this in ppt! Take it with a pinch of salt, but qualitatively it’s correct!

So looking back at Florian’s picture, for the blue waves to have been caused by Florian’s ferry, there are two options:

A) they would have to have wave speeds faster than the ferry’s bow wave and wake

B) the ferry would have had to come from the direction of the island, so that the waves propagated in that deeper channel behind the sand bank before the ferry made its way around the sandbank.

Option A is impossible, because wakes travel at maximum wave speed (similar to a sonic boom in the atmosphere, where sound is travelling at maximum speed, forming a cone with the air plane at its tip, only here it’s a 2D version, a V-shaped wake with the ship at its tip). So if the wake is traveling at maximum speed already, then the blue waves can’t go faster than that.

For option B, looking Florian’s ferry up on a map, I saw that that ferry goes around a small island, which is the land you see in his picture. But a quick glance at the map shows that even though the sand bank seems to end where the ship would have had to have gone in order to create those waves, the island is still very much in the way. So this can’t be the solution, either.

This map is published at http://map.openseamap.org under a CC BY-SA 2.0 license

So let’s take another good look at the original picture.

Remember those wave crests that I marked in blue? Well, upon closer inspection it turns out that they are tidal gullys and not wave crests! (Which is what Florian confirmed when I asked whether he remembered the situation) Guess I have been barking up the wrong tree all this time!

So back to the wave crests that I marked in red:

What we see here is exactly the depth dependence of the phase speed that I plotted above. Right at the sand bank, the water is shallowest and waves are slowed down (we see that both in the green wave crests that seem to be falling back and start breaking as they get closer to the sand bank [both indicating that the water is getting shallower], and in the red wave crests right at the sand bank). But as the water gets deeper again on the far side of the sand bank (which depth measurements in the map above seem to confirm), the phase speed picks up again (as it has to — see my plot above) and the wave crests accelerate again. Hence we have the weird phenomenon of waves suddenly speeding up!

Very long explanation, I know, but still pretty cool now that we solved it, right? I love #friendlywaves — if you have any mystery wave pictures, please do send them my way! :-)

# A #friendlywaves from Lofoten

A #friendlywaves post: you send me the pictures, I talk about physics! Today: My friend A sent me these lovely pictures from Lofoten, knowing I love wave watching. And there is so much to see!

Let’s begin with the picture above, where we are looking out over the stern of a ship towards a bridge. There are two different kind of things that jump out to me: The ship’s wake and the tidal current.

The ship’s wake consists of two parts: The turbulent wake we see right in the middle of the picture, behind the A-frame crane (in between the red lines below), and the feathery V-shaped wake (some of the individual “feathers” are marked in green).

And then there is the turbulent backwater behind the bridge’s pylons that’s caused by the tidal current going through underneath the bridge. Pretty cool, isn’t it?

And now on to the next picture, that is one of the most beautiful wave pictures I’ve seen the last couple of weeks: Now we are sailing in the wake of a second ship.

We are following the other ship a bit off to the side, therefore the perspective is a little confusing. Between the red lines, we see the other ship’s turbulent wake. Additionally, it has an interesting V-shaped wake that actually consists of two stacked Vs, a bit like this: <<

One of the Vs is the actual bow wave radiating from where the ship’s bow cuts through the water, the second one detaches further backwards from the ship. Both Vs are marked in dark green below. But to the left of the picture, in light green, I marked some of the individual “feathers”, wavelets that make up the V-shaped wake.

Isn’t it fascinating? I love this.

# Concave sea wall focuses waves into interesting pattern

Early morning Kiel fjord — today even featuring a hot-air balloon!

But, more interestingly, the wake of this police boat. I find it already pretty cool in the picture below: The fjord is calm and mirror-like, but inside the ship’s V-shaped wake the surface changes completely and the reflections look totally different (now only reflecting the sky back, not the cranes). And, of course, the V-shaped wake itself has quite a large amplitude, too.

A little while later, the wake has not only reached the sea wall, it is already being reflected back away from the wall. See the original wake at the bottom of the picture below, and the reflection further away, near the five bollards?

Looking slightly further right, we see the concave shape of the sea wall here, and how waves are being focussed similarly to how radio waves are focussed towards the receiver with parabolic antennas.

So as the reflected waves propagate out further and further, they little by little reach a focal point.

Which you see in the picture below: An area of higher waves in the middle of the water, seemingly for no reason.

And the area where waves interfere and amplitudes are so high moves a bit over time, but it’s a quite persistent pattern.

Had I just come across this pattern without seeing it develop, I don’t think I would have been able to explain what is going on here.

And see how, now that the wake has passed, the checkerboard pattern of interfering waves in the foreground is a lot more prominent again?

# Observing the shape of shallow water waves on a beach

When I go wave watching back home in Kiel, I tend to be drawn to other places than this particular one, but sometimes you find gold in the places you least expect it. And then, what is usually a really nice and easy to explain kind of wave — shallow water waves — acts up and becomes difficult. You win some, you lose some…

In these pictures, we see these very weirdly-shaped waves in very shallow water. When the water is shallow enough, waves don’t even have a “proper wave shape” any more, they look more like pool noodles that are being pushed towards the beach, but in a very inefficient, sideways manner.

When the water is shallow enough, the waves also change their behaviour in that usually we can just add two incoming wave fields and get a good idea of what the resulting wave field will look like, but in very shallow water, things become very nonlinear and messy.

Do you see how wave crests seem to start clinging together, resulting in weird X and Y shapes?

Usually we can easily calculate the velocity of shallow water waves just from the water depth, but here in this picture, all kinds of weird and wonderful things are happening and I don’t actually have a clue what’s going on.

A little bit further along, some other surprising wave watching: A wake is running onto the shallow beach.

What i find really interesting here is the one long, straight wave crest that runs all the way across the picture, and then how different the waves look on either side of the draining pipe (actually, I don’t know what’s going into Kiel fjord there, although now I am curious…).

On the left side of the picture, the wave crest hits the shore and that’s the end of waves on that side. On the right side, though, wave crests continue on in shallow water for quite some time before they end up on the shore, and here we see how they get bent by the changing water depth (Remember? Phase velocity of shallow water waves should only depend on water depth, so the deeper the water, the faster the wave). The wave crests get slowed down a lot faster on towards the shallow water on the left than towards the deeper water on the right, thus the right side starts overtaking the left, bending the whole wave crest around.

# Walk to & visit at GEOMAR as wave watching occasion

These very long and regular wave crests were caused by a ship — they are one side of the V of a ship’s wake that’s about to reach the sea wall here.

I love how well defined the wave crests are — on one side you see the reflection of the bollards and the bright sky, on the other side the reflection of the dark sea wall.

Below you can see the waves approaching the sea wall and then their reflection:

And I realized that it’s a very good thing I didn’t own a smartphone when I was still working at GEOMAR, otherwise you’d be completely fed up by these kinds of pictures by now :-)

For the rest of this post: More pictures of more wakes as seen from the roof top terrace, which very nicely show you the V-shape…

I just love how you see the “feathers” that, together, make up the V-shaped wake.

And several wakes on top of each other…

And below you see the turbulent wake in the middle of the V as a lighter line where the surface appears smooth due to turbulence.

# Wake watching at Ratzeburger Segelschule

The waves that ducks (or ships, or anything else moving through water) make are called “wakes”. And wake watching is so much fun! A wake is pretty much the Mach cone in 2D — waves from the duck can’t propagate faster than the leading edge of the wake. When a wake passes by, that’s the moment you would hear the supersonic boom of an aircraft in the Mach cone analogy. That explains why the wake develops as a feathery V with the duck at its tip. Or several ducks at the tips at several Vs, as in the picture above. How awesome is that?

But wake watching is usually more complicated than just Vs with stuff at their tips. See for example below — two ducks with Vs, two row boats also with Vs, but with the additional eddies where their oars pushed through the water.

See how different the two pairs of wakes look?

Or here, we have wakes again, but they are now occurring on a lake surface that shows more different waves: The half circles that my feet tapping on the water made! And, if you look closely, you can spot algae growing in the lake in the foreground of the picture.

Isn’t it fascinating how you see the algae in the foreground, but the further you look towards the other shore of the lake, the more you see the reflection of the sky or the other shore instead?

That’s due to a phenomenon called “total internal reflection“: For light that hits the interface between two different materials (air and water in this case), there are angles at which it can cross the interface (at steep angles we can look from the air into the water and see the algae), and others where it can’t (at small angles, we cannot look into the water, light gets reflected at the lake’s surface and we see the opposite shore instead). This fascinates me time and time again!

Note how the duck in the image above doesn’t actually have a wake? That’s because it’s swimming too slowly for the wake to develop — it is just radiating ring waves in all directions.

Below, this is kind of a boarder line wake — we can see the V developing, but se also still see the ring waves in the spot where the duck first started swimming.

But of course, wakes are only straight Vs when the ship is driving in a straight line, too, otherwise it will get deformed like below.

And just because I love this picture so much: Here is a wake coming across the lake

# A #friendlywaves from Tampa, Florida

Anyone who might be new to my blog because of yesterday’s presentation at #SiPManc — please don’t be scared and run away, this is the most complicated #friendlywaves I have ever gotten, usually things are A LOT easier! :-)

I love #friendlywaves! Victor sent me the picture above. He took it in 2017 in Tampa, Florida, and I think it’s so fascinating! There is so much going on, let’s try to make sense of it!

First, the most obvious thing making waves here: The two boats. Clearly they are making waves, and they might explain a lot of what we see here. But on the other hand, they might not.

Below, you see the part of the wave field that is 100% due to these two ships: Their V-shaped wakes (in red) and the turbulent wake behind one of the ships (in yellow).

The very prominent wave pattern (marked in red in the image below) might be due to these two ships as was suggested to me, but if it is, then those ships changed course quite drastically before they created the waves I marked in the previous picture (and I can see no evidence of such a change of course, usually a turn would leave a trace similar to this one).

If the boats, as I assume they did, came out from underneath the bridge and sailed in a more or less straight line (and that seems to be the case judging from their wakes as indicated in the picture above), there is no way they could have made waves that travel in front of their V-shaped wake. Similarly to how you can’t hear the supersonic aircraft before the supersonic boom (because the sound can’t travel faster than the speed of sound and the pressure signal thus gets formed into the Mach cone), waves can’t outrun their wake (which is like their 2D Mach cone). So I don’t believe that those waves were made by those two ships. Rather, I believe that they were made by a ship that is no longer visible in the area we are able to see.

So remember, this is the wave pattern we are trying to explain (Marked is only one wave crest, but you see that there are several parallel to the marked one):

We do nicely see how the wave is reflected by the straight sea walls. But what direction is it traveling in? And what caused it? Let’s speculate!

First: let’s consider the very weird shape of the body of water shown in the picture. Quick search for Tampa on Google Maps lets me believe is that the picture was taken more or less from the position of the white star and the view is the area between the two red lines. Looking at that map, we see that the water we see opens up into four different water ways: One to the north, one to the east, one to the south east, and one to the south west. The two to the south eventually open up into Tampa Bay.

The wave field that we are trying to explain would look somewhat similar to what I drew in below (green):

My best explanation of that green wave field above is this: A boat that went on the course that I drew in in yellow:

So far, so good. Wanna know why I believe this is what happened? Then this is the picture for you!

Assuming the boat followed along the yellow track, the other lines are the wake it would have produced:

• green: Those are two parts of the wave field that I marked above that I am fairly confident of: The wake propagated across the body of water, got reflected and came then over towards the photographer. Note how not all waves reach the shoreline close to the photographer yet? That’s because they are the “newer” waves that haven’t traveled for long enough to reach that spot
• light blue: The “newest” waves that aren’t very long yet and are traveling in an area where we can’t clearly make out the presence or absence, let alone direction, of waves. They are fanning away from the “green waves” because the ship is turning (similar to here).
• dark blue: Those is a part from the wake that originated on the other side of the ship, got reflected, and now traveled across the body of water to reach the point where the picture was taken from. They do so at an angle that looks like they might be reflections of the incoming green waves (which is another possibility which I can’t rule out with 100% certainty). Newer wakes from that side, once they’ve been reflected on the shore, will lead to waves almost parallel to the green part of the wake and would be indistinguishable from those in the picture.
• orange: Those are “old” wakes that must have happened when the ship came out of that inlet, but that would not interfere with our picture because their reflection stays caught within the inlet itself.

This is the best explanation of what must have happened that I can come up with, and I have thought about this quite some time (more on that at the end of this post) :-)

But then there are tons of shorter wave length waves that we have to explain, too: See those marked in red, yellow and green below.

I am confident that the ones I marked in red are wind-driven waves coming across the open area. Their direction also agrees quite well with the wind directions the flags indicate (marked with a white arrow above). I believe that the ones I marked in yellow and in green are two separate wave fields at a slight angle, but that might be an optical illusion, I am not quite sure.

If we go back to the map, I believe the wave fields I marked above would look pretty similar to the ones I drew in below (I changed the red waves above to magenta waves below, because red was already taken. Note the wind direction marked with a white arrow: it looks pretty much perpendicular to the now-magenta wave crests):

And looking at the angles in that depiction of the waves, I could imagine that the green wave field is a reflection of the magenta wave field where that one hits the shore on the side where the picture was taken from (see light blue wave crests). As for the yellow one: I still have no idea what caused that. But maybe there need to be some mysteries left to life? ;-)

To end on something that I am confident in: The half circles near the bottom of the picture are the result of something (two buoys? two small boats?) moored on that pier, bobbing up and down in the waves, thus radiating wave rings with shorter wavelengths and higher frequency than the wave that is exciting the movement.

But after all this hard work (more on that at the bottom of this post) — let’s take a minute and look at those beautiful interference pattern again where the wave fields cross each other and create a checkerboard pattern. How amazing is this?

Phew! I love #friendlywaves, but this was quite a challenge! How did I do, Victor? :-)

If you or anyone else have any comments or suggestions — I would love to chat about alternative explanations!

P.S.: Just to give you an idea of what my process was like: It involved late night scribbles on a tea bag (because that was the best “paper” I had available on my bedside table in the hotel in Manchester) and I needed to play scenarios through in my head…

…and some sketches on my phone while I was on a train…

This is how much I love wave watching! :-)

# Nena sent me some #friendlywaves from Lago Maggiore

This is a #friendlywaves challenge, where I try to explain other people’s wave photos and they tell me how I did.

I love it when my friends see waves, think of me, whip out their cameras, take pictures, and send them to me! In this case, Nena even used a telephoto lens and took the amazing pictures below that she allowed me to share with you!

They are the perfect example for talking about wakes when a ship doesn’t just go straight ahead. Because, of cause, ships going straight ahead are the easiest case, like the one we see below.

Picture by Nena Weiler, used with permission

Here, we see the two different constituents of a wake: The turbulent wake that is the white stripe right behind the boat, that turns blue a little way behind the boat but stays a lighter color than the surrounding water.

And then there is the V-shaped wake with the boat at its tip. This V-shaped wake consists of very many individual waves that are fairly short in the direction parallel to their crests, and that are shifted slightly so the further away from the boat you look, the wider the V opens. I usually call this the “feathery” wake, since it consists of all these little “feathers”, but since I need the “feather” image for something else today, I’ll just call it the V-shaped wake here.

Now when the boat takes a turn, this messes up the structures of the waves making up the V-shaped wake (or makes them more interesting, depending on your point of view). Below, the boat has taken a right turn, which you can see from the turbulent wake that starts right behind the boat as a white stripe that then changes color to a lighter blue than the surrounding water (with a darker stripe to each side, and then the V further out).

Picture by Nena Weiler, used with permission

Now looking at the individual waves of the V-shaped wake, we see that they get bunched up on the right side of the boat’s trajectory, while they are getting fanned out on the left side.

Now imagine the boat’s trajectory as the shaft of a feather. If you have ever bent a feather, you will have observed that on the side the shaft is bent towards, the individual barbs (I looked this up: barbs are the little thingies that spread outwards from the feather’s shaft) get bunched together, while on the other side they fan open.

So far, so good. Still with me?

Now what happens as time goes on is that the V opens up — the two sides move away from each other. We don’t usually notice this because we are used to focussing on the wake relative to the ship rather than to some fixed vantage point. But if we looked at a fixed point while a ship going past, we’ll see the wake spreading over time until one side of the V reaches us.

Picture by Nena Weiler, used with permission

And this spreading of the V is what’s making interpretation of the picture below a little difficult. The picture below is showing almost the same part of the ocean as the one above (see the little white and blue moored boats in the bottom right corner of the lower picture? They are the same boats that are visible at the left of the bottom right corner above), only a little later. During the time between the two pictures, the ship moved further towards the bottom left corner, but also the wake spread further apart.

Above, you see that some “barbs” start running into each other (the ones where the bend is strongest, where there is foam on breaking waves because the waves suddenly become a lot steeper due to interference). So some time later, they have grown longer and are now crossing each other, which leads to the checkerboard pattern located right inside the bend of the boat’s trajectory. If you follow the V-shaped wake from the boat backwards, you can still make it out, even though it’s been deformed by the ship turning around.

Picture by Nena Weiler, used with permission

Tell me, Nena, is your family happy with this explanation? :-)