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.
The spectral theory is essential to study wave phenomena. A practical objective is to optimize the transmission in a waveguide, even if there are some perturbations inside. It is known that for certain frequencies, there is no reflection by the perturbations but it is not a priori clear how to find these frequencies. Anne-Sophie uses complex analysis for that. The idea is to complexify the (originally real) coordinates by analytic extension. It is a classic idea for resonances that she adapts to the problem of transmission. This mathematical method of complex scaling is linked to the method of perfectly matched layers in numerics. It is used to solve problems set in unbounded domains on a computer by finite elements. Thanks to the complex scaling she can solve a problem in a bounded domain which reproduces the same behavior as in the infinite domain. Finally, Anne-Sophie is able to get numerically a complex spectrum of frequencies, related to the quality of the transmission in a perturbed waveguide. The imaginary part of the complex quantity gives an indication of the quality of the transmission in the waveguide. The closer to the real axis the better the transmission.