Freak Waves

We were yesterday at the Café scientifique in Headingley to give a talk about the mysterious freak waves.

In average, two large (>100T) ships sink in the ocean every week. This huge amount of loss comes from various reasons, such as bad weather conditions, collisions or pirates’ attacks. But among the events that cause ships to sink, one is particularly terrifying: freak waves. Considered as legendary for centuries, these sudden extreme waves started to raise the scientists’ curiosity 20 years ago.

Munchen

On December 1978, the safest ship in the world, called the München, started his journey from Europe to America. But on the night of the 12th of December, she suddenly disappeared in the middle of the Atlantic ocean. Two months later, the life-boat was recovered, and the investigators concluded that a huge force had struck the ship 20m above the water level. But what this force was, was a mystery.

However the mariners had an idea about what could have hit the München at 20m height.  For centuries, a lot of them came back from the ocean explaining that an unexpected huge wave, a sudden wall of water, had damaged their boat. They all described it  the same way: “a single breaking wave of the size of the tower block”. From Washington Irving, even Christopher Columbus would have faced such a wave. 

children3However, from the scientists’ point of view, this kind of sudden extreme waves couldn’t exist. In the 1990’s, Dr. Jim Gunson from the Met office used an analogy with children in a classroom to explain why such waves should not occur: in a classroom, some children are taller, some are smaller, but they all average around the same size. The chance to find a child who is three or four times taller than the others is very low.

The same happens with waves: from the linear model used at that time, extreme waves such as those described by the mariners should occur only once every 10 000 years. So scientists did not believe that waves faced by the mariners were any kind of physical event, but assumed it was simply the consequence of bad weather conditions.

But on the 1st of January 1995, an extreme wave was recorded on the Draupner offshore platform, in North sea. While all the waves measured about 12m height, one suddenly hit the structure with a height of 26m, which is more that twice the height of the waves around. From that moment, scientists started to study these waves, and a definition was given: freak waves are waves at least twice as height as the waves around. 

Since then, many other events have been recorded and freak waves are a real threat for ships.  While actual boats are designed to resist waves hitting them with a force of up to 30T/m2, the force applied by freak waves is estimated to be up to 100T/m2, which is equivalent to 20 locomotives on one meter square! This obviously causes many damage to boats, and is therefore a big concern for the shipping industry, as well as a threat for mariners.

Our PhD projects aim to develop a mathematical model of these waves, and understand how to generate them at a given position in a basin, in order to test their impact on models of ships or wind turbines. This will then help the mechanical engineer to design those structures better so that they can resist the load and stress applied by such extreme waves.

Note: Most of this article was inspired from the great documentary “Freak waves” published by the BBC in 2002. You can find it on that link: Freak Waves

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Maths and coastal defence.

As a result of ongoing coastal erosion and storms, coastal land-based infrastructure is in constant danger of both large-scale damage and flooding. Existing coastal defences, predominantly vertical and/or planar walls, although providing some protection, are frequently breached in severe conditions, which breach necessitates costly repair and/or rescue operations. Such walls are particularly ineffectual in deflecting the high-amplitude waves that occur during storms and/or high tides. The video below illustrates this problem, in which water can be seen breaching the wall and inundating the area behind it.

For videos of coast ripped up by waves, see, e.g., https://www.youtube.com/watch?v=mFrRXlulVFU

For videos of coast ripped up by waves, see, e.g., this link.

Applied mathematical modelling can play a crucial role in dealing with this problem. It enables one to determine the shape of the sea wall that is optimal on two fronts: first, which prevents breaching by water; second, which minmises the load applied by the incoming waves on the wall.
Usual vertical wall, as shown in the first movie.

Usual vertical wall, used in the first movie.

New shape, defined as optimal using mathematical methods.

Optimal shape, defined using mathematical methods.

Using mathematical and computational optimisation methods, a new barrier shape has been designed that dramatically changes the behaviour of the wave impacting upon the wall: mathematics has played a key role in protecting the land-based infrastructure from the waves! The next video illustrates the results of this new design by demonstrating how the breaching has been minimised (by comparison with the first video).

blog_coast2

Note: the shape has been designed by students William Booker, Thomas Goodfellow and Jacob van Alwon.

What is a standing wave ?

We’ve previously talked about standing waves. But what is it ?

Visually, it looks like a dancing wave: the wave is going up and down, but does not seem to travel, because each point has a constant amplitude.

Standing wave generated from a wave maker.

Standing wave generated from a wave maker.

From the video above, you can see that the wave is not moving the same way everywhere: some parts of it go up and down with a maximum amplitude (these points are called antinodes) while some other points seem static (these are called nodes). This is characteristic of a standing wave.

Well, that sounds cool, but how does it occur ?

A standing wave is formed when two identical waves (same frequency, amplitude, speed…) travel in opposite directions and meet: the sum of these two waves will create a standing wave. This is how we generated it in the movie above:  by setting an appropriate wave maker frequency, we can generate a wave which propagates along the basin. When it reaches the end wall, it is reflected and comes back in the opposite direction. When meeting the incoming waves, standing waves appear.

Formation of the standing wave : the wave is generated by the wave maker and propagates along the basin. When it reaches the wall, it is reflected and comes back in the opposite direction. When meeting the incoming waves, standing waves appear.

Formation of the standing wave: the wave travels along the basin, is reflected, and meets the incoming wave to give a standing wave.

The number of standing waves in the basin depends on the frequency of the wave maker. This one must satisfy an equation involving the length of the basin, the number of waves that we wish (the “wave number“) and the wave length (basically the distance between two crests). In the video below you can count 7 waves, but we could have generated 1, 2 or 13 waves just by changing the frequency of the wave maker.

Can this occur in daily life ?

Of course it can ! Just take a string, hold it in both ends, and move it up and down. Give it a try with different speeds: one of them (corresponding to the resonance frequency), will create a standing wave.

You can also observe standing waves in shallow water, in music (harmonics), light, and a lot of other applications.

Ok, but how is Maths involved in this ?

Well, you can use Maths to simulate these standing waves: by solving waves equations analytically or numerically and choosing an initial condition (that is, the solution at initial time t=0) that satisfies the characteristics of a standing wave, you can end up with the kind of simulations below. These simulations can then be used to set testbeds, which are a primordial step in the production process of any object. For instance,  we can predict the behaviour of a boat in such waves, in order to improve its design.

Simulations of standing waves.

Simulations of standing waves.

 Nice, right ? 😉

School of Maths open day

DSC_0069Saturday 12th September was the School of Maths open day in Leeds!

We showed experiments of waves breaking on a beach, waves impact on a wind turbine (or should we say “on a toy looking as a wind turbine”), coastal defence, and also standing waves (if you do not know what a standing wave is, then don’t miss our article about it…coming soon!). The students could also see our simulations of waves breaking on a beach and those of standing waves.

We’ll post videos of these experiments soon, and explain how important it is to have reliable mathematical models of these different situations!

Hello world!

Tomasz

Tomasz

Floriane

Floriane


We are two PhD students from the University of Leeds, working on mathematical modelling of sea waves. Why are we interested in sea waves? What are we exactly doing? What is it needed for?


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We’ll show you the fun behind Maths and how you can study water waves without getting your hands wet!