## Waveguides – Modellansatz

H. Wind

Everybody has experimented with resonating frequencies in a bathtub filled with water. These resonant eigenfrequencies are eigenvalues of some operator which models the flow behavior of the water. Eigenvalue problems are better known for matrices. For wave problems, we have to study eigenvalue problems in infinite dimension. Like the eigenvalues for a finite dimensional matrix the spectral theory gives access to intrinsic properties of the operator and the corresponding wave phenomena.

Anne-Sophie Bonnet-BenDhia from ENSTA in Paris is in conversation with Gudrun Thäter about transmission properties in perturbed waveguides.  This is the third of three conversations recorded during the Conference on Mathematics of Wave Phenomena July 23-27, 2018 in Karlsruhe for the Modellansatz Podcast. Anne-Sophie is interested in wave guides: Optical fibers that can guide optical waves while wind instruments are guides for acoustic waves. Electromagnetic waveguides also have important applications.

## Pattern Formation

If one puts a pan with a layer of oil on the hot oven in order to heat it up one observes different flow patterns over time: In the beginning it is easy to see that the oil is at rest and not moving at all. But if one waits long enough the still layer breaks up into small cells which makes it more difficult to see the bottom clearly. This is due to the fact that the oil starts to move in circular patterns in these cells. In our example the temperature difference between bottom and top of the oil gets bigger as the pan is heating up. For a while the viscosity and the weight of the oil keep it still. But if the temperature difference is too big it is easier to redistribute the different temperature levels with the help of convection of the oil.

This means that the system has more than one solution and depending on physical parameters one solution is stable while the others are unstable. Mariana Haragus, Professor in Besançon at the University of Franche-Comté, is doing research on this important question for engineers as well as mathematicians.

Gudrun Thäter was in conversation with her in the context of the Modellansatz Podcast about Bernard-Rayleigh problems: Where do these convection cells evolve in theory in order to keep processes on either side of the switch? This had been one of the interesting research topics at our 2018 Conference on Mathematics of Wave Phenomena.

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