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Pré-Publication, Document De Travail Année : 2005

Resonance modes in a 1D medium with two purely absorbing boundaries: calculation methods, orthogonality and completeness

Résumé

Studying the problem of wave propagation in media with absorbing boundaries 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, 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 without any approximation using three methods. The first one is the straightforward use of the closed-form solution in the frequency domain, then the residue calculus. Then 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 and completeness of the complex-valued resonance modes is investigated, leading to the expression of a particular scalar product. The last method is the expansion in biorthogonal modes in the frequency domain, the modes having eigenfrequencies depending on the frequency. Results of the three methods generalize or/and correct some results already existing in the literature, and exhibit the particular difficulty of the treatment of the constant mode.
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Dates et versions

hal-00003357 , version 1 (26-11-2004)
hal-00003357 , version 2 (25-04-2005)
hal-00003357 , version 3 (01-03-2006)

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Jean Kergomard, Vincent Debut, Denis Matignon. Resonance modes in a 1D medium with two purely absorbing boundaries: calculation methods, orthogonality and completeness. 2005. ⟨hal-00003357v2⟩
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