“In a storm, you first have a chaotic situation with waves of different wavelengths”, he says, and describes how this tends to be followed by a more structured storm phase where the longer waves travel first and the shorter waves follow.
He thinks it is interesting to study the three-dimensional structure of waves from a mathematical perspective, i.e. not only wave height and direction of travel, but also the waves from a sideways angle. This third dimension plays an important role in the stability of the waves, or how much disturbance they can stand. Stable waves can travel further than unstable waves.
Erik Wahlén explains that he has previously worked on mathematical models for both breaking waves and waves in fast-flowing water. His ongoing research also includes a new project about waves in fast flowing-water and in water of varying density, i.e. the weight of the water relative to its volume. His research may in the future also have links to ‘monster waves’.
Monster waves are an unusual but well-known phenomenon out at sea, for example off the coast of South Africa. These high waves can appear without warning and put sailors’ lives at risk. Erik Wahlén points out that it is uncertain how they come about – it may be due to ocean currents that slow the waves, making them higher, or it may be instability in the waves. Mathematical models of waves are a step on the way to finding answers to why monster waves form.
However, storms and monster waves seem distant today. The surface of the sea in Lomma Bay is calm; only small waves break here and there. What does Erik Wahlén see with his mathematical eye? He points out to the waves and explains that, when the sea is calm, they can almost be regarded as linear waves. They are so low that a simple mathematical model would be sufficient to calculate them, i.e. a linear calculation.
The difference between linear and non-linear calculations is really quite easy to understand. If two small waves of the same size are regarded as linear then they can basically be added together to form a wave of double the height. However, if the waves are higher, it is not possible to add them together mathematically and get a simple solution in the form of a wave of double the height. The larger the waves are, the more they affect one another when they collide, which complicates the chain of events considerably.
“Really, all ocean waves are non-linear, both large and small”, stresses Erik Wahlén, saying that the more advanced methods of the non-linear model must therefore be used to calculate what happens when waves collide.
When Erik Wahlén stands on the beach looking out at the sea, he sees a surface covered with non-linear waves that also have three-dimensional challenges. But that’s not all. In order to make the description mathematically correct, the waves can in fact be regarded as calculations of differential equations.
A differential equation is a mathematical method of describing a change. A partial differential equation is a slightly more complex variant that is used when multiple factors affect the change in question, for example when the change is affected by both time and place.
“All water waves are described using partial differential equations”, says Erik Wahlén.
So, on your trips to the beach this summer, you can sit back with a refreshing ice cream in your hand and enjoy the knowledge that the sea before your feet is a rippling illustration of non-linear partial differential equations.
Text and photo: Lena Björk Blixt
