Resonance modes in a 1D medium with purely absorbing boundaries
Résumé
Studying the problem of wave propagation in dissipative media can be made by searching for "resonance modes" or free oscillations regimes. In the present article, a simple case is investigated, which allows to enlighten the respective interest of different, classical methods, without any approximation, some of them being rather delicate. This case is the 1D propagation in a homogeneous medium having two purely resistive terminations, the calculation of the Green function being done by using three methods. The first one is the straightforward use of the "compact" solution in the frequency domain, then the rules of residues. The second one is the expansion in orthogonal modes in the frequency domain, the modes having eigenfrequencies depending on the frequency. Calculating the derivative of the eigenfrequencies with respect to frequency allows to get the same result than the first method. Finally the method of separation of variables (space and time) leads to a solution depending on the initial conditions, and can be applied to the Green function. The question of the orthogonality of the (complex) resonance modes is investigated with several points of view. The results of the three methods generalize or/and correct some results already existing in the literature. Finally the extension of the problem to a medium where dissipation is coupled to propagation is investigated.