# What Causes Giant Rogue Waves?

## Introduction

Sailors have spun yarns for centuries about gigantic rogue waves that could suddenly come out of nowhere to capsize the ships of unwary mariners. Scientists didn’t believe them because the stories seemed at odds with everything else known about waves. Then cameras and other instruments began to capture undeniable proof of the existence of rogue waves. Ton van den Bremer, an expert in fluid mechanics, talks with Steven Strogatz about what science has learned about how rogue waves form, whether it’s possible to predict them and how the waves can be recreated in a lab.

Listen on Apple Podcasts, Spotify, Google Podcasts, Stitcher, TuneIn or your favorite podcasting app, or you can stream it from *Quanta*.

**Transcript**

**Steven Strogatz** (00:03): I’m Steve Strogatz, and this is *The Joy of Why*, a podcast from *Quanta Magazine* that takes you into some of the biggest unanswered questions in math and science today. In this episode, we’re going to ask what causes the monster waves in the ocean known as rogue waves.

(00:20) Throughout human times, the oceans have swirled with all sorts of legends. Think of the Flying Dutchman ghost ship that disappeared during a storm in the 1600s and is now forced to sail the seas forever. Or the reptilian sea serpents that fishermen swear have menacingly popped up from the surface of the water, or the Sirens of Greek mythology who lured sailors to their deaths with their soothing, seductive songs. We know all of these to be myths. But there’s one mystery of the sea that is not a myth, and it can be deadly: rogue waves. These are giant waves that seemingly come out of nowhere. They can slam into ships or hit oil platforms. And because the ocean is so big, with so many factors, they’re really hard to study because they’re difficult to actually witness.

(01:08) One of the most famous is called the Draupner wave. It struck the Draupner gas pipeline platform in the North Sea in 1995, reaching an astounding maximum height of 25.6 meters, or 84 feet. That’s about the size of a six- to eight-story building. It was the first time a rogue wave was ever measured with instruments.

(01:30) Capturing a rogue wave in the vast ocean is rare, so we still know relatively little about them. But scientists like Ton van den Bremer are trying to change that. Dr. van den Bremer uses wave pools and modeling to study rogue waves at Delft University of Technology in the Netherlands. He’s an associate professor of civil engineering and geosciences, and also a senior research fellow at the department of engineering science at the University of Oxford. Ton, thanks so much for joining us today to talk about rogue waves.

**Ton van den Bremer** (02:01): Thank you. It’s a pleasure.

**Strogatz** (02:02): Well, I’m really looking forward to this. It’s a fascinating subject. Let us start with just the basic issue of characterizing them. What makes a rogue wave rogue? Like, how is it different from the ordinary ocean waves that we see at the beach? Or tell us a little more about how big they are. How fast they can travel.

**van den Bremer** (02:19): Usually, you have a lot of waves, right? So you can compare one to the next. And that’s indeed what you do. So you look at basically a characterization of what we call the sea state. So this is an average of how tall the waves are. And then rogue waves are defined to be much, much greater than what is the average wave in that sea state. And so we actually very specifically say that a rogue wave is defined as being two times a quantity called a “significant wave height.” And significant wave height is basically a measure of how big the waves are at that point in time. And if your wave exceeds two times that, we say, “Oh, that is now a rogue wave.”

**Strogatz** (02:55): So that’s interesting. It’s a kind of outlier relative to whatever…

**van den Bremer** (02:59): You could think of it maybe as something, as an abnormal wave, both in a mathematical, precise sense — we have the normal distribution — but also in a sort of common parlance, right? We have the normal distribution of waves, and this is something that’s very far away from normal. So much bigger than what you would expect.

(03:14) But of course, you have to be careful with that because what you expect depends on how long you wait. So waves, they come and go, right? They occur all the time. So that means if you wait for long enough, you always have a rogue wave. It’s just a question about how long are you willing to wait. And rogue waves are basically waves you have to wait for a long period of time.

**Strogatz** (03:34): What was that phrase you used earlier? The characteristic wave height?

**van den Bremer** (03:37): We call this the significant wave height. In any type of statistical description, you have the standard deviation, right? It’s a width of — if I have a sample, I’d like to describe that sample, say, of the height of a human being, right? I have a sense of what’s the mean height, but I also have a sense of the variance of that population. And in a sense of the variance, or the variation of the surface, is the significant wave height, how large the waves typically are. It’s called a significant wave height.

**Strogatz** (04:03): Why was it so impossible for scientists in the olden days to believe the sailors’ tales about rogue waves?

**van den Bremer** (04:08): What you have to imagine is that, you know, we’ve been studying the sky and, indeed, outer space for some time, and it’s been relatively easy. But the ocean is much harder. We can only study this from ships, or from wave buoys or from the coast. And if you study things from the coast, you know, you only learn things about the immediate vicinity. So everything coming from, basically, from the deep, deep ocean, the surface of the deep ocean, has come from ships. So that has been sailors’ tales.

(04:33) And it’s incredibly difficult, of course, to take a measurement from a moving object. You have a ship, And you have to take a measurement. So all of that has been sort of anecdotal. And it hasn’t really been until the start of the ‘70s and the ‘80s when the oil exploration started, and we started to build out in the deep ocean on platforms — so, fixed structures from which we could reliably measure things. And that is indeed also how the Draupner wave was measured. It was measured off the Draupner platform, a place in the Norwegian North Sea where you would not normally be taking measurements. But now you have an outboard platform. And that gives you an opportunity to take these measurements.

(05:07) This field is a field that people have been predicting these or speculating that these had been happening, but there hadn’t been really any data until this point in the ‘90s. Really recent, if you like. At which point we finally had a measurement where we said, “OK, this is now not a measurement error.” Previously, we had perhaps measured rogue waves, but we thought, “OK, maybe the instrument was broken, right? Surely it can’t have been this high. It’s too big, there must have been something wrong.” And now, this was a measurement that was reliable. It was reliable, because it was measured properly. And also there was some damage on the oil platform at that same height. So the wave actually hit the oil platform. So there was more than one source of evidence. And this made it the first credible rogue wave.

**Strogatz** (05:46): Let me picture it a little. I tried to give a visual image by saying six- to eight-story building. Is that the right way for me to picture it, or what would you normally tell someone?

**van den Bremer** (05:56): I think that’s a good description. That magnitude, I mean, that gives you an indication. But you have to imagine that, basically, you’ll have a number of buildings that are much, much less tall, coming your way all the time, right? These are the normal waves that you see coming at you. So you were quite used to, say, seeing two- to three-stories tall. And then all of a sudden, there’s one which is double or three times higher than that. And that is indeed what we’re talking about.

**Strogatz** (06:19): What do we know about the conditions that would cause such a giant wave to form? Do we know, like, what has to happen in the ocean or the atmosphere?

**van den Bremer** (06:26) There’s a lot of different mechanisms, and because we only have so few measurements, it’s very difficult for us to tell which is the dominant mechanism. But there have been two types of mechanism that, if you like, have been competing for the prize. And one of these is simply — we call this linear dispersive focusing. And I’ll explain. Basically, the ocean consists of many, many waves. And they travel in … not random but in lots of directions. And there are lots of these different waves with all the different amplitudes. And at some point in time, quite a few waves come together, and they build up. They kind of (what we say) linearly superimpose. So there’s one wave of a meter and other wave of another meter, and it all adds up. And if you wait for long enough, you’ll find that lots of waves add up. And that’s one mechanism.

(07:11): So scientists find this less exciting, maybe because it’s a completely linear mechanism. So it’s just linear superposition, you’ve got to wait for long enough. And that probably explains quite a lot of the rogue waves out in the ocean, but not all, because there’s a second mechanism that has to do with nonlinear effects.

(07:26) So it’s an effect where these waves that come together, they now just don’t add up linearly. So it’s not one meter plus one meter giving two meters, but it’s one meter plus one meter all of a sudden giving three meters. And that nonlinear effect is probably the reason this is such an attractive field, because that nonlinear mechanism occurs not only in ocean wave physics, but it occurs in very many different systems. And it’s described by some nonlinear Schrödinger equations that we see in lots of applications, including optics, including ocean waves. And it leads to basically one and one no longer adding up and giving rise to bigger waves.

(08:01) Now for that second mechanism, that can certainly lead to rogue waves. But it’s also restricted to the really deep ocean, so it can’t be close to the coast. And for the Draupner wave, the water wasn’t deep enough for that to have played a role there. So it has played a role in some rogue waves, but not in the majority.

(08:18) And then there are lots of other mechanisms, mechanisms that are quite poorly understood, like the effect of the wind blowing really hard, bursts of wind. There’s also other effects associated with rapid changes in the water depth. So all of a sudden, you have a step in the water depths. The water depth changes. That can also lead to rogue waves.

**Strogatz** (08:38): So should I visualize — whether through the linear addition mechanism or the nonlinear Schrödinger mechanism — should I picture a solitary wave? Is it one big hump of water? Or should I think of a wave train of two or three of these giants coming at me?

**van den Bremer** (08:53): Now that’s a good question. So typically, waves in the deep ocean are dispersive. That means that if you have a certain wave that has a certain shape, that changes as it travels. So it’s unlike sound waves, where we can reasonably easily communicate with one another because all the waves that we make when we speak to each other kind of arrive at our ears at the same time. Now, that’s not generally the case for ocean waves. And so when we have these extreme waves — indeed, they come in groups or in packets, so it’s a few of them. And depending on the sea state, there can be fewer or more waves, but typically, it’s between two and, say, five to six waves that make up this thing called a wave packet or a wave group. And one of these will be the biggest one, and then there’ll be another few big ones, and then it will be gone.

(09:41) So the story of it coming out of nowhere is perhaps not true, but it is coming fast. Because these waves, they have periods, meaning every wave takes about somewhere between six and 20-30 seconds. So that’s the sort of order of magnitude. So that it comes quickly and it goes quickly, and the group takes a little bit longer. But it’s all gone on the scale of minutes.

**Strogatz** (10:04): So when you refer to these — did you say something like 20-30 seconds for a period?

**van den Bremer** (10:09) That can be true. It depends on where you are, it depends on, basically, where you are in the ocean, and how long the wind has had time to input energy into the ocean. So if you’re in the deep ocean, you get longer periods of, say, upwards of 10 seconds. But if you’re close to the coastal environment, you have wind waves that have been blowing, you get shorter periods. So younger seas where the wind has only been blowing for shorter time. So the wind is just, you know, the storm has just started, you start by seeing relatively quick waves, meaning short periods. And then over time, you get to longer periods.

**Strogatz** (10:40): Just to make sure I’m following because I think I know what period means, like, say in a first-year physics course, where I’m picturing a mass on a spring bobbing up and down in a simple harmonic oscillator. So here in the context of a wave, the period would be if I pictured myself sitting on the beach, a wave has come in and it’s hit my sandcastle. And then the amount of time till the next wave even further demolishes my sandcastle, that’s the period.

**van den Bremer** (11:03): Exactly. And then you have, of those waves coming to demolish your sandcastle, right? The rogue wave might be accompanied by, maybe, either side, like a couple other waves that are a little bit less tall. But together, they come as a group.

**Strogatz** (11:16): Uh huh, that’s your group picture. Good. And then there was the term “dispersive,” which we know in ordinary language: If you tell a crowd to disperse, then all the individuals spread out. What does it mean for a wave or a wave packet to disperse?

**van den Bremer** (11:30): To — it means — and this is also basically the mechanism, this first mechanism I described behind rogue waves — it means that waves of different periods, together with at different wavelengths, they travel at different speeds. And that allows — if I make a few different waves, which have not only different amplitudes, but more importantly, different frequencies, that they can catch up with one another. Right. And that allows for a mechanism where I can make different waves (meaning different periods) and I can let them go. And that means if I make them at the same time and I let them go, they all travel at different speeds, so they disperse.

(12:05) But it can also be the other way round: If they’re made a different times, then they can come together. And that leads to then the big wave, because the faster traveling waves are just catching up with a slowly traveling waves. And that leads to these tall waves. And that is indeed, if you like, half of the mechanism of linear focusing. Different period waves traveling at different speeds and they catch up and they lead to a big wave.

**Strogatz** (12:26): I see. So because — I noticed you used the word “focusing” earlier. So that’s what was in your mind. Or no?

**van den Bremer** (12:32): Two aspects, right? So that we call “focusing,” basically that arises from different periods, right? So it just focuses in one direction if you like, right? Because they’re different waves, different speed, and they all catch up — they focus, they kind of meet each other. But of course, you can also have something we call directional focusing. And this is where we have waves now not in one direction, but multiple directions. And they — that’s much more easy to visualize, right? You’re driving from different directions, and you’re all coming into one point. And that we call directional focusing. So there’s these two aspects that are focusing that together make focusing happen.

**Strogatz** (13:06): Very helpful. One last thing to mention about this. If I think of a giant wave, my mind immediately goes back to that famous woodblock Japanese print, “The Great Wave Off Kanagawa.” If it were real, would it be a tsunami? Is it a rogue wave? And also, what’s the distinction between those?

## Introduction

**van den Bremer** (13:24): So I think it is mostly an artwork and it’s very difficult — OK. It’s — I don’t know the year, but it’s some time ago. So we don’t know, we didn’t have — like, we had the Draupner platform, where we were able to record. In this instance, we didn’t really have a recording. It probably is — given that it was a picture, it probably is a wave close to the coast, because otherwise it probably wouldn’t have been recorded. This sort of rules it out in terms of being a traditional rogue wave, because we tend to have one of those mechanisms — at least in terms of the traditional rogue wave mechanism, the nonlinear mechanism — that doesn’t really occur close to the coast. So it’s probably not one of the nonlinear rogue waves. And what it really is, it probably wasn’t a tsunami, because it — a tsunami would be so long, it probably wouldn’t fit on this woodcarving. I think it was a woodcarving, it probably is. You know, the wavelengths are of the order of the size of the painting.

(14:13) So it could have been, it could have been a rogue wave, but we have no way of telling because we’d have to compare it to all the other waves at that time. So it was mostly an appealing illustration. And the only thing we as scientists when we took off our scientific hat and we become artists for a period of time, is that we saw some motion, some wave-type behavior that we also observed in our experiments of the Draupner wave, this Draupner wave that we reproduced.

(14:42) And this kind of brings us to another topic, and that’s wave breaking, right? So I’ve described this idea that waves can meet each other and that they can sort of superimpose, and if you wait for long enough, the wave will always become taller and taller and taller. But there is a limit to this, right? At some point, the wave will break. And we’re all familiar with this because we see waves break, and we see the white-capping, we see the splash. But what it means is that typically, traditionally, in a wave that we used to think about, it’s also the limit of how tall the wave can be. It sets a maximum, so no waves could be bigger than that.

(15:13) And what we showed in this work of modeling this Draupner wave, is that had the Draupner wave consisted of wave components, individual waves that were all traveling in one direction — we call this unidirectional — then it probably would have broken. It wouldn’t have become as tall as you’ve described. But if we then made it out of multidirectional wave, or indeed specifically a crossing system, where we had wave energy from two directions, it was only then that we could get a much taller wave. And that much taller wave could become as tall as it was measured in 1995. And part of that motion we saw of this breaking, that became different.

(15:51) So rather than this rolling breaking that you see if you have unidirectional waves, if you go to the beach, you tend to see this sort of rolling-forwards motion. If we have crossing waves, then we have a splashing upwards. And that splashing upwards appeared in this woodcarving of The Great Wave. And we — again, with our scientific hat off and our artist’s hat on — we saw what we were able to reproduce in a lab. And what we’re able to use to explain the Draupner rogue wave.

**Strogatz** (16:19): Oh, that’s, that’s fascinating. So when you mentioned rolling waves, I’m immediately reminded of, of surfers. You know, I mean, of course, we see little waves when we’re sitting at the beach on a calm day. But if you look at the competitions in Hawaii, of the surfers inside the waves as the wave is rolling over the top of them — well, at least that’s the image I have in mind when you’re speaking about rolling waves.

**van den Bremer** (16:41): That’s why I used it. It’s a, it’s a familiar image people know, because they’ve been to the beach, but even in the deep ocean, right? If you look out of an airplane, you’d look down on a relatively windy day, you see bits of white everywhere. And there are bits of waves that have kind of fallen over, that if you like have become too steep. So they’re so steep that they can’t really contain themselves, if you like, and they just fall over. And you see a bit of splashing. And it’s that sort of bit of splashing that you see when you study surfers and you see at the beach. But it also happens in the deep ocean, very far away from any surfers.

**Strogatz** (17:12): Well, let me go back to the scientific point that you were making, though, to see if I got that. That you say if the waves were unidirectional, all more or less moving in one direction, once they start getting combined to make a big wave, the point was they will be limited in their height. You claim there should be some multidirectional mechanism.

**van den Bremer** (17:32): Basically, if the wave is becoming too steep — and this is easiest to think about in one direction, right? You’re just trying to sort of walk up and down a hill. If that hill is too steep, then you fall off it. And that’s what happens to the unidirectional picture, right? This traditional idea of something that’s simply too steep. And what happens is, basically, the wave overtakes the fluid. The fluid goes faster than the wave, and it sort of falls over.

(17:53) But if you have crossing, you can imagine a different mechanism, because now you have two waves coming from two different directions. And when they meet, there is no obvious falling-over direction anymore because there is no obvious direction in which the wave is going. And so what happens instead is that the motion goes upwards. You might imagine if you run into each other, the only way you can avoid each other at that point is to both jump upwards. And in loose terms, that’s sort of what’s happening, we think.

(18:19) And somehow the fact that we have this upward behavior causes the wave to actually become steeper than it would be. Otherwise it would have sort of fallen over. And it now has become steeper and taller. And therefore basically this Draupner recording which we have, we have a recording of from 1995 of the wave at the time, which was really, really steep. And we show that it probably would have broken long before reaching that steepness if it had been energy coming from one direction.

(18:45) And we should add, of course, we’ve got a recording of this wave. But we’ve got a recording at one point. So we have one point — a laser basically pointing down and it’s recorded the surface. What we haven’t got is information as to where the waves have come from. So we kind of had to infer, looking back after all those years, what could have been the directional composition and make some assumptions there. And they’re showing that there is evidence to suggest that maybe there was indeed what we call crossing behavior going on.

**Strogatz** (19:11): Let’s make the transition into your own laboratory here. It’s — you’ve mentioned it a little bit so far. And I hope we can go deeper into it. It sounds like you’ve done some kind of reenactment or simulation or something that’s qualitatively trying to get at what was happening in this Draupner wave. Am I hearing you right?

**van den Bremer** (19:30): That’s absolutely correct. This field — because it’s very difficult to create out there in the ocean waves you want, right? In fact, you cannot. You have to wait for the wind. So we make them ourselves. So we have what are swimming pools, large swimming pools, but the sides of the swimming pools are wave paddles. So they’re devices that allow us to control exactly the input signal, and thereby we make basically the waves in the way we like. And traditionally these pools are either very long — we call them flumes — with one or two wave makers at one on end, or they’re big square pools that have wave makers on one end. And this makes it very difficult to make these crossing scenarios where waves are traveling from multiple directions.

(20:11) These experiments were done at the FloWave Ocean Energy Research Facility at the University of Edinburgh. And that facility was quite unique because it’s completely round. So it’s a round pool. And all of the side walls consist of individual wave makers. So there are 164 individual wave makers around this pool. And we can control them in any way we like. So we can make waves coming from different directions.

(20:33) What we did is, we simply tried to make the Draupner wave. And we did so where we had all the energy in one direction, and then we split it up in two systems at an angle to each other. And then we varied that angle. And that way, we learned basically that — we were able to study whether we could create this Draupner wave in a laboratory environment. And we showed that if we were in a unidirectional environment, we sent all the waves in one direction, we simply couldn’t make it. It required this directionality, which we could make in this facility.

**Strogatz** (21:01): I love the sound of this. I want to hear a lot more about this. So let me see. The round pool, how big of a — should I picture an Olympic-sized swimming pool but rounder?

**van den Bremer** (21:13): Right, I think there are a few of these things around the world. And some indeed exceed the size of Olympic swimming pools. This one is only 25 meters in diameter. So relatively short. Two meters deep.

**Strogatz** (21:23): And a wave maker. Let me understand what that is. So there’s a paddle that has some kind of motor attached to it, that can pump it?

**van den Bremer** (21:30): No, not pump it. It’s just a paddle, right? So it’s just — so basically, over this two-meter distance, the wall is on a hinge. And it sort of flaps. And it flaps in a very controllable way with a motor with the control mechanism. And we can just specify how fast it flaps, which sets the frequency of the wave, and how large is the flapping motion, its amplitude. That sets the amplitude of the wave.

**Strogatz** (21:52): And you say there are 164 of them, I suppose more or less equally spaced around this?

**van den Bremer** (21:57): No, it’s continuous. So the whole wall, basically —

**Strogatz**: Oh, continuous.

**van den Bremer** (22:01): They’re all — they’re all — they’re all, everything, there’s only wave maker, basically.

**Strogatz** (22:06): The whole wall is made of wave makers, but you said 164 of them.

**van den Bremer** (22:09): Yeah. So this gives you — you can divide 360 degrees by that number, and I’m sure you get some other number. And it just means a few degrees, basically is the resolution in the directional distribution that we can achieve in this way.

**Strogatz** (22:21): And so then the game is to try to make the — are you shooting for the biggest wave you can make in that pool?

**van den Bremer** (22:26): That is a game one can play. And indeed we have. But that’s just a slightly boring game. Because it turns out, one can make enormous waves if you just completely focus things axisymmetrically. So you make something completely round, and indeed, we’ve shown that you can make these axisymmetric waves. And these axisymmetric waves, they occur in multiple applications in nature. For example, if you drop something through a free surface, like a coin, you’ll find there’s a little “bloop” noise. And that is a little axisymmetric wave that actually is forming. You will find it in Faraday waves. And you can also find it here. And it leads to this sort of spike, this real big spike. So you might imagine if you send waves in from all 360 degrees, they all kind of add up and superimpose, and it just leads to an enormous spike.

(23:09) In the real ocean, that never occurs. But there’s elements of this behavior. Elements of this, what I described as, you know, two cars that are meeting head on or that crash into each other. Imagine having 360 of them, right? All driving at each other, towards the center of a bull’s eye, if you like. That leads to a really big wave.

(23:27) That, we think, is giving some insight into the mechanism that occurs in the real ocean. Not really when waves come from all directions, because that never happens, except for maybe in hurricane conditions, when you have a rapidly changing hurricane. Then the waves come from one direction in one minute. And then a little bit later, they come from another direction. But there always is something we call a directional distribution. So some spread of the energy over multiple different directions.

**Strogatz** (23:48): And so when you are trying to simulate conditions that would have led to the Draupner wave, about how many different directions were in play? Just two or three or…?

**van den Bremer** (23:57): First of all, what we had to do, we couldn’t just generate any wave. So we tried to make sure that at the center of this facility, at what we call the central gauge, basically in the middle of the bull’s eye point — at that point, we wanted to reproduce the measurement time series from the Draupner platform scaled to the lab. So we actually had, like the recording in 1995, at that central location, had exactly what was observed over not just at that wave, but a few waves around it. So some time. And then we had some freedom over the directional information, but we wanted it to be realistic. But then we had to make some assumptions. And there was some evidence from the weather at the time, “hindcasts” of the weather — so, after-the-fact predictions of what the weather was like — that there was some information that there were wind and systems from different directions.

(24:42) So we had some arguments to say that all of the energy present at the time was probably coming out of two different systems. And those two different systems, we gave directional spreading of their own. So if you like, there were two systems which each had some directional spreading. So they were over a couple of angles. And when these two systems had a different angle between them, in that angle between the two systems, that was the angle we varied.

**Strogatz** (25:04): Give me a rough idea. What, is there a kind of resonance effect? Is there a best angle?

**van den Bremer** (25:09): We basically found that if everything was in one direction, it would break, right? It would break. And then we had to separate the systems. And we found that at 60 degrees, we were getting close. And we had to go up to 120 degrees. But we didn’t do this at huge resolution. So whether it’s 100 degrees, or 140 degrees, that’s difficult to say, but at least anything greater than 60 degrees gives you enough spreading of the energy, enough crossing, enough of a crossing angle, that we can get waves tall enough. And if we make the crossing angle even bigger, I think bigger waves were possible. So it just gave an indication that crossing was necessary. But we didn’t dare say how much crossing exactly.

**Strogatz** (25:47): OK. And you did briefly mention the idea of scaling. And so maybe we should underline that for a minute, that you’re not trying to get the same wave as what was seen in the ocean.

**van den Bremer** (25:58): Or you might imagine it’s — the real wave is tens of meters, right, in the ocean, or 25 meters, whereas these experiments, we’re talking tens of centimeters, so it’s much smaller. So basically, what we have to do is, you have to think about, I think, the most important part is the steepness, right? So the steepness is a slope. So it’s not the height, but it’s the height divided by the length over which the variation happens. And that we can scale. That’s one part. So we have the shape right, right? it’s got the right shape, the right angles, if you like. And then we know the governing equations that describe this system. And these governing equations allow us to also not only scale the shape, but once we’ve scaled the shape, also adjust the timescales. Because we also have to adjust the timescales, because our waves in the lab are much quicker, much shorter periods than our waves in the field.

(26:47) And the physics that drives these waves allows us to scale this based on something called a Froude number, a characteristic number that describes the system. And that scaling has been successful. And that scaling has been the reason that a lot of ocean wave science has been done in labs because it works so incredibly well. And you can de-test ships — test wave conditions in the laboratory environment — without losing all of the key processes. In fact, retaining many of the key processes required to get things right.

**Strogatz** (27:16): Well, let’s shift gears a little bit to talk about the statistics of rogue waves. I believe one of the studies that you were involved in looked at the graphs of the statistics and asked questions about what would happen if you removed just a little bit of — say, 1% of the energy from the high frequency tail of those graphs. Do you want to tell us about that study?

**van den Bremer** (27:36): So this is all about, again, about trying to make things in the lab. And if you think about — I have to introduce a concept here called the wave spectrum. So if you want to understand the wave climate — so this is basically what the waves are like at that point in time — you plot something called the wave spectrum. So it gives you an indication of the distribution of the energies over all the different frequencies, right? You have your typically — this wave spectrum has a certain peak, and that’s where the dominant, you know, that’s probably the period you’ll observe when you’re sat out there on the beach. But there are also a tail. And the tail is on the right of the graph. So at the high frequencies, there’s a tail, and that that tail has energy in it. But that tail is incredibly difficult to make in the lab. Because what you have to do, for example, if you have a 1 Hertz wave, to get to the tail, you might need to get to 5, 6 Hz. And it’s that tail that is out there in the ocean. But it’s difficult to get that tail right in the lab. And this is where some of the scaling kind of breaks down.

(28:33) And we show that if you get that tail wrong — typically because your wave maker, your paddles simply can’t generate these high frequencies — these frequencies get so high, because of the scaling much higher than in the real ocean — they get so high that your paddle typically can’t vibrate quick enough, right? And it just stops working. And therefore that tail is very hard to make. And therefore people typically cut it off. They say, “Well, above 2 Hz, typically we’re not going to make it because we don’t trust our wave generation.”

(29:00) But then there are nonlinear processes in wave physics that create this tail, because the tail needs to be there for the spectrum to be in equilibrium with the nonlinear equations that describe these, for example, the nonlinear Schrödinger equation. And if you take out the tail, it might just come back. And in coming back, it creates bigger waves.

(29:19) So our paper sort of studied is that you have to be incredibly careful in the lab, if you’re interested in rogue waves, especially when they’re, when they’re unidirectional (in one direction), if you accidentally take something out, using no answer, there’s not much happening in that tail, there’s very, very little energy. When it comes to rogue waves, you might just accidentally be removing something that the nonlinear physics put back in, but more than you wanted as you get more waves. And that’s not really representative. So you have to be careful, as we showed, about this tail. It’s very important, and it’s difficult to scale and to get right in the lab.

**Strogatz** (29:51): So should we think of it as really a technical point about the difficulty of doing these sorts of experiments faithfully?

**van den Bremer** (30:00): I think you can. And I think it will always be a limitation, because we’re probably never going to be able to make these, really, vibrating a heavy paddle really, really quickly at a range of frequencies. That’s always difficult. If you have a musical instrument, you can make some frequencies really well and others not so well. And that’s just driven by the size of the instrument, in this case, the size of the paddles.

(30:18) So I think it’s basically a cautionary tale about some of the experiments. And there are a lot of things being designed based on wave tank experiments, where previously we thought about the tail as indeed a technical assumption: “We just make a practical assumption, we set the limit at 2 Hz. And it’s not important. Let’s ignore it.” And we show that it is, in fact, far more important than we even thought. And we should be very, very careful about it.

**Strogatz** (30:41): Let’s pan back a little bit to a broader view of this. You mentioned other fields — where say, the nonlinear Schrödinger equation comes up. In optics; maybe we could think about… People may be aware that, you know, nowadays, with high-speed internet, some of the communications may be happening over optical fibers. So are there other fields where waves come into play? And there are counterparts of rogue waves that we could be applying this knowledge to?

**van den Bremer** (31:07): That’s right. Well, I should immediately qualify, those are not my field. So I’m really, you know, going out on a limb. But I’ll try and say that, indeed, it’s in those fields you’ve described, where you also have, basically, optics we’re dealing with also with waves, right? The equations describing this are similar in their reduced form.

(31:24) So many of these systems basically combine two things. There is dispersion, which is this mechanism I had described before: different waves of different dispersive behaviors. When you and I talk, we don’t have so much dispersion. But when we have optical waves, we might have dispersion. When we have — indeed water waves, we have dispersion, that’s the one effect. And then the other effect is the effect of nonlinearity. Basically, one and one not adding up to [two]. And these two basic effects occur, basically, across so many wave phenomena.

(31:54) And if you take them together, and you try and write the simplest possible equation that has both, you have this nonlinear Schrödinger equation. And this equation is powerful because it occurs in all these different fields simply because it’s the simplest way of getting dispersion and nonlinearity. Taking that together. The form the nonlinear Schrödinger equation takes — the weight that’s given to dispersion on the one hand, and nonlinearity on the other hand — is different. It depends on the field, depends on the properties of your material, say, if you talk about an optical cable, versus a basin of 25 meters by one meter full of water, so the coefficients are different. But the system, the equation is the same.

**Strogatz** (32:35): Forgive me for poking you, since you say it’s not really your area. But is there anything, do you know, from your friends who work in optics, do they see anything like rogue waves that they need to be concerned with?

**van den Bremer** (32:46): This nonlinear Schrödinger equation and variants of it have — have all sorts of beautiful solutions. And some of these beautiful solutions are examples of what could be rogue waves. And these beautiful solutions have certain mathematical forms. And some of these — we could almost call them “creatures” — types of solutions that, like the solitary wave you described, but also things called “breathers.” These are waves that, they breathe. So they become large, and then they’re gone again. And they can breathe in space — they can occur once in space — or they can occur in time, they occur once in time. And these breather structures, they, you know, we can make them in a lab really easily in our fluids lab, right, our big water channel.

(33:25) But similarly, these breathers — and there are many of them — have been observed in optics and in different media. And indeed, that’s where I think the fields meet, that these creatures, I like to call them, or breathers, they’ve been observed in a number of different media, a number of different types of matter including, in my case, just a really simple thing called water.

**Strogatz** (33:48): Very nicely put. What about math and computers? We haven’t talked about them too much. I mean, I know from being a person who — myself, I like to work on nonlinear systems of various kinds — that they are very challenging to analyze mathematically. And I’m sure even more so for you where you have this spatial aspect and some multidimensions need to come into play. The computations must be hard, the math must be even harder. Do you have a few words for us about all of that? Is it strictly an experimental and statistical subject?

**van den Bremer** (34:21): No, it is — I think, originally, I guess — I mean, in terms of rogue waves, rogue waves were studied mathematically, perhaps before they were studied much in the field, because we weren’t taking many measurements. And the study of these waves in the lab has only really been taking off… I guess there were mathematical community starting water waves, perhaps because they are relatively simple equations. You might think of them as hard because there’s evolution in space and time, but at least water is water, right? So its properties don’t change. And it’s hard because there is a free surface, right? There is basically air, and at some point there is water. Where the dividing line lies between the two has to be defined. That makes it hard but this is an incredibly rich field. And we’re fortunate, I think, because the governing equations are simple, the boundary conditions are hard. And this makes for something where you can make progress and can apply a rich set of mathematical tools.

(35:14) But if you think about the frontiers in this field, the frontiers are probably wave breaking. And this is exactly when those equations stop being so simple. So when our beautiful mathematical tools that allow us to understand exactly how waves propagate, they stop working. And there, we have to really resort to large computers and do brute force numerical simulations.

(35:35) Imagine a breaking wave. And you would have seen one of those at the beach, and you see all the whitecap and the air entrainment. And that becomes incredibly difficult. And that, I think, is where the, where the frontier sits in this field. And that’s also where colleagues and I are looking at alternatives. For example, machine learning-type techniques, where we make this combination of, on the one hand, a nice mathematical equation for the nonbreaking waves, an equation has been around for decades. And we combine that with laboratory experiments where we measure lots of breaking waves, and lots of different locations. And we combine those bits of information together with data-driven machine learning-type techniques. So we can still have the power of our simple (or I say “simple”) nonlinear Schrödinger-type equations that have this nonlinearity and dispersion, and then have these new, or relatively new, computational methods, machine learning-type methods to understand what I think is the final frontier, wave breaking, to be able to predict a wave evolution, even those big waves that start to break.

**Strogatz** (36:35): Oh, well, that’s a beautiful summary. And very inspiring to think about the role that machine learning could play, because it does seem like such a hot area in practically every branch of science, and even outside of science. So I guess I’m not totally surprised to hear you say it. But still, it’s very, very exciting. And you’ve touched on this question of prediction. That was my, the last question I wanted to throw at you. Do you think we’ll ever be able to predict rogue waves like in the same way that people aspire to predict tornadoes or earthquakes? I mean, in every field, this is a tricky business.

**van den Bremer** (37:10): I think there are two levels of prediction here, right? One is predicting environments in which the probability is increased. So I think that we’ll be able to do. So, for example, if we have a certain sea, or a certain shipping route, and we say, “Under these conditions, it will be far more likely that they arise.” And that’s probably the level we need because a rogue wave — it occurs, you have to wait for a couple of waves. And if you just — if the frequency becomes unacceptable, right? So you say, “They occur once every half a day on average. So, once every couple of thousands of waves.” Then it becomes unacceptable to use this as navigation. That level we can probably get to. And we’re making some progress, probably based on our understanding of the field data.

(37:51) The other type of prediction, where we are at one location, we sort of have a live prediction — “In one or two seconds, we’ll be hit by a wave?” or indeed… And that’s very difficult. And I think there are applications where this is necessary. For example, landing a helicopter on a ship. This sort of live forecasting is necessary, you have to sort of say with a few seconds delay, “OK, this is going to be the wave system. This is going to be what the waves are like. This is going to be what your ship motion will be like.” And that is incredibly difficult when it comes to rogue waves simply because they are these statistical outliers. So you either need huge amounts of data to predict this. Basically, you need huge amounts of data to be able to predict these things because they’re so, so rare.

**Strogatz** (38:31): This has been a great pleasure. Thank you very much. We’ve been talking with wave expert Ton van den Bremer. Thanks again for joining us today.

**van den Bremer** (38:37): Thank you. It was a pleasure.

**Announcer** (38:43): Space travel depends on clever math. Find unexplored solar systems in *Quanta Magazine*’s new daily math game, Hyperjumps. Hyperjumps challenges you to find simple number combinations to get your rocket from one exoplanet to the next. Spoiler alert: There’s always more than one way to win. Test your astral arithmetic at hyperjumps.quantamagazine.org.

**Strogatz** (39:14): *The Joy of Why* is a podcast from *Quanta Magazine*, an editorially independent publication supported by the Simons Foundation. Funding decisions by the Simons Foundation have no influence on the selection of topics, guests or other editorial decisions in this podcast or in *Quanta Magazine*. *The Joy of Why* is produced by Susan Valot and Polly Stryker. Our editors are John Rennie and Thomas Lin, with support by Matt Carlstrom, Annie Melchor and Zach Savitsky [*and Nona McKenna*]. Our theme music was composed by Richie Johnson. Julian Lin came up with the podcast name. The episode art is by Peter Greenwood and our logo is by Jaki King. Special thanks to Bert Odom-Reed at the Cornell Broadcast Studios. I’m your host, Steve Strogatz. If you have any questions or comments for us, please email us at [email protected]. Thanks for listening.