Abstract:

We consider here acoustic waveguides with a bounded cross-section, where  only a finite number of modes can propagate at a given frequency, other modes being evanescent. If the waveguide is locally perturbed, an incident wave will generally produce a superposition of reflected propagating modes. As a consequence, only a part of the total incident energy will be transmitted from the inlet to the outlet. But at some exceptional frequencies and for particular incident waves, it may occur that 100% of the energy is transmitted, the only effect  in reflection being a superposition of evanescent modes in the vicinity of the perturbation. An important question for the applications is to identify these reflection-less frequencies.

We show that these frequencies can indeed be  computed as discrete eigenvalues of a non-standard eigenvalue problem. The main idea is to select ingoing  waves in the inlet and outgoing waves in the outlet by applying a complex scaling in the inlet and in the outlet, with two different parameters, with imaginary parts of opposite signs.  The resulting spectrum contains, in addition to the real reflection-less frequencies, complex eigenvalues that can be used to quantify the quality of the transmission.

Form the mathematical point of view, the spectral problem is non-selfadjoint and includes a continuous spectrum. For particular configurations, it has the so-called PT-symmetry, with interesting consequences.

Several numerical results  illustrating the  variety of configurations that can be analyzed by this approach will be presented and discussed.