Hello, my name is Peter Clarkson. I'm Professor of Mathematics here at the University. My research is in the study of nonlinear differential equations, in particular their applications to things such as wave phenomena, which is what I'm going to talk about today. Pick the three types of wave phenomena, rogue waves, tsunamis and solitons. The soliton has a history that dates back to the 19th century. They were first observed by John Scott Russell in 1834 whilst riding on horseback beside the narrow Union canal near Edinburgh. He described his observations as follows: 'I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses when the boat suddenly stopped – not so the mass of water in the channel which it had put in motion; it accumulates around the brow of the vessel in the state of violent agitation, then suddenly leaving it behind, rolled forward
with great velocity assuming the form of a large solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course along the channel, apparently without change of form or diminution of speed. I followed it on horseback and overtook it still rolling at a rate of some eight or nine miles an hour preserving cylindrical figure some
30 feet long and a foot to a foot and a half in height. Its height gradually diminished and after a chase of one or two miles are lost in the windings of the channel. Such in the month of 1834 was my first trance interview with that rare and beautiful phenomenon which Ihave called the wave of translation.
Subsequently Russell did extensive experiments in the laboratory wave tank to study this phenomenon. Study Russell observed solitary waves which are waves of permanent form, hence he deduced they exist which is his most significant result. Missiles showed
that the speed of propagation of the solitary wave was related to the height of the of the channel, the amplitude of the wave and the force due to gravity. I was lucky enough in 1995 to witness the recreation of the soliton in the same Edinburgh canal near Heriot-Watt University. Now the canal goes over the Edinburgh ring road in a viaduct. What happened was a boat came along and suddenly stopped just before the viaduct started and it created this wave which then went across the viaduct like that. This was the generation of a soliton wave.
Some other scientists undertook to study this phenomenon too, particular Boussinesq and Rayleigh in the 19th century and they obtained descriptions of the solitary wave given by Scott Russell. Subsequently, in 1895 two Dutch scientists Korteweg and de Vries derived an equation governing long one-dimensional small amplitude gravity waves propagating in a shallow channel of water and derived this equation which actually had appeared in the work of Boussinesq in the 1870s. In 1955 Fermi, Pasta, Ulam and Tsingou studied a problem involving what are called nonlinear springs which are springs that have an interaction with their nearest neighbour. They studied this numerically and found that if they set it in a certain format to begin with they'd have some oscillations and then the same original format would reappear. Interestingly, the work was done by the four of them but the paper was just written by three, the three men. In 1965 two American scientists Norman Zabusky and Martin Kruskal determined to study numerically the phenomena observed by Fermi Pasta, Ulam and Tsingou and they did this by looking at the Korteweg, de Vries equation derived by the Dutch scientists in 1895. And what they found was the discovery of what they call solitons, these two solitary wave solutions of the equation they found passed through each other as though nothing had happened. You'd normally expect two waves when they interact to just dissipate, you wouldn't expect them to reappear as though nothing had happened. They did this as follows: they took the Korteweg, de Vries equation and the red line is the initial condition they started with. As time progresses it goes towards the blue line which you expect then you're going to have wave breaking, this height getting much larger and then suddenly collapsing. Instead the wave broke into these eight waves which all lined up in order of height. What they had shown was that the Korteweg, de Vries equation has this solution where the speed of the wave is proportional to the amplitude and therefore taller waves travel faster than shorter waves. So what happens is the taller wave comes along interacts with a shorter wave and overtakes it and they go on as though nothing had happened. This phenomenon has been seen in the ocean in various places. Here is a mathematical model of a soliton that has got a curved shape and here is a picture showing an application of such a thing in the Straits of Gibraltar. You see these curved shapes and they're all in form of height just as predicted by Zabrusky and Kruskal. The taller wave leads and then they get that shorter as you go along. What you don't realise from this picture is these are not actually waves on the ocean surface they're internal waves as the ocean is made up of layers of fluid and these waves can travel in the intersection between the different layers of the fluid. In fact they can be very powerful waves travelling in such a phenomenon. Another application shows the interaction of two waves in shallow water. This photo was taken of the coast of Oregon, you see these two waves, you see this interaction just exactly as the mathematics will actually predict of what will actually happen.
Now I want to talk about broke waves. These are isolated structures within usually high amplitude such as the ocean wave in this 1834 woodcut Fuji by the Sea by Hokusai Mariners had reported such waves on the ocean but people didn't believe them. That was not until 1995, when the wave height measurement was observed on the 1st of January in 1995 at the drachma oil rig in the north sea off the coast of Norway. So this is just a framework the height of the wave and you see this one very large wave in the middle and that is the rogue wave. It's a lot higher than all the other waves, it just suddenly came out of nowhere and then disappears. This is the first time that we have proof that these existed. Subsequently, rogue waves has appeared in various other contexts, such as optical fibres, Bose-Einstein condensates waves in superfluids, atmospheric waves and also in finance. Here is an example from the optical fibres you can see the wave travelling along and suddenly it is much larger than all the others. So you get to see, this is what a rogue wave is, it suddenly just rises from great height and just then disappears.
The final topic I want to mention is tsunamis. What do you do if you're at the seaside and you notice the sea gradually withdrawing and the water getting further and further away further than ordinary tides. Well, sadly everybody knows the answer today, run like the clappers up the nearest hill or higher ground if you can. As our ancestors might have put it thousands of years ago the sea god has breathed in and you better make yourself scarce before he breathes out again. The tsunami that brought the world's attention was the Boxing Day tsunami in 2004 of Sumatra. It was such a devastating effect the people in Thailand didn't stand a chance because it got there within 30 minutes. In Sri Lanka and India the wave took 5 hours to get there and it was 8 hours to get to Somalia. Tsunami waves are generated relatively often from a variety of sources, however serious tsunamis, that is from the point of view of the loss of life, take place less often. And a challenge is to find a credible role that mathematicians can play in predicting their danger or in alleviating their impact. So potential contributions predicting an earthquake this is a grand challenge problem presently not possible. Where it is possible for mathematicians to contribute to the problem of tsunamis it's the following: the modelling of tsunami wave generation and their propagation across the ocean, the design of early warning systems or some components thereof and the clarification of the character of tsunami waves. So what happens in a tsunami, you have this situation before the earthquake and then the ocean plates start moving and depending on how the earthquake happens, whether the ocean floor goes down or whether it goes up depends on whether you get a crest or a trough of the leading part of the wave. So you can understand what is going to happen purely by the effect on the ocean floor. Then the wave travels across the ocean and its relatively low height until it gets towards the coast. As it approaches the coast the front of the wave starts getting slower but not the back of the wave, so the wave run will rise up and they create the sheer height of the wave and if it's a depression that created the wave, you get the trough first and that's when the water will come back from the beach and you see the mass of water has been created behind it and that's causing the power. In the 2004 tsunami the height was estimated one to two feet and the length of the wave was about 50 miles and so that means that the wave was traveling speed have 450 miles an hour the speed of a jet aeroplane. So the 50-mile wave will pass by in about seven minutes it's how it's going fast in the open ocean but near the shore it slows down and this creates the power of the wave. Now tsunamis are no taller than normal wind wave but then they're much more dangerous. In the normal waves the waves go round in circles and they can come and go without flooding higher areas. The tsunami wave it travels in a straight line and has a much bigger volume of water. So to summarise, water waves with horizontal scales much longer than the local ocean, like tsunamis, travel with an approximate speed related to the height and the gravity – the same formula that Scott Russell had in 1834. And the shape of the wave that reaches the shore depends on how the earthquake changed the ocean floor and how far this tsunami has propagated. Finally, I'd like to quote from Paul Dirac: 'the mathematician plays a game which he himself invents the rules while the physicist plays a game in which the rules are provided by nature but as time goes on it becomes increasingly evident that the rules which the mathematician finds interesting are the same as those which Nature has chosen.' Thank you