Abstract. In shallow-water waveguides a propagating field can be decomposed in three kinds of modes: the propagating modes, the radiating modes and the evanescent modes. In this paper we consider the propagation of a wave in a randomly perturbed waveguide and we analyze the coupling between these three kinds of modes using an asymptotic analysis based on a separation of scales technique. Then, we derive the asymptotic form of the distribution of the mode amplitudes and the coupled power equation for propagating modes. From this equation, we show that the total energy carried by the propagating modes decreases exponentially with the size of the random section and we give an expression of the decay rate. Moreover, we show that the mean propagating mode powers converge to the solution of a diffusion equation in the limit of a large number of propagating modes.Key words. Acoustic waveguides, random media, asymptotic analysis.AMS subject classifications. 76B15, 35Q99, 60F05. IntroductionAcoustic wave propagation in shallow-water waveguides has been studied for a long time because of its numerous domains of applications. One of the most important applications is submarine detection with active or passive sonars, but it can also be used in underwater communication, mines or archaeological artifacts detection, and to study the ocean's structure or ocean biology. Shallow-waters are complicated media because they have indices of refraction with spatial and time dependences. However, the sound speed in water, which is about 1500 m/s, is sufficiently large with respect to the motions of water masses to consider this medium as being time independent. Moreover, the presence of spatial inhomogeneities in the water produces a mode coupling which can induce significant effects over large propagation distances.In shallow-water waveguides the transverse section can be represented as a semiinfinite interval (see Figure 1.1) and then a wave field can be decomposed into three kinds of modes: the propagating modes which propagate over long distances, the evanescent modes which decrease exponentially with the propagation distance, and the radiating modes representing modes which penetrate under the bottom of the water. The main purpose of this paper is to analyze how the propagating mode powers are affected by the radiating and evanescent modes. This analysis is carried out using an asymptotic analysis based on a separation of scale technique, where the wavelength and the correlation lengths of the inhomogeneities, which are of the same order, are small compared to the propagation distance. Moreover, the relative fluctuations of the medium parameters are small on the scale of the square root of the wave length over the propagation distance. This is the interesting scaling regime corresponding to propagating modes and where the coupling via the environment gives a strong mode coupling. In the terminology of [7] this is the so-called weakly heterogeneous regime.Wave propagation in random waveguides with a bounded cross-section and Dirich...

International audienceCet article présente lʼétude asymptotique de la densité dʼénergie de la solution de lʼéquation de Schrödinger ayant un potentiel aléatoire à décorrélations lentes. On montre que la transformée de Wigner de la solution de lʼéquation de Schrödinger aléatoire converge en probabilité vers la solution dʼune équation de transport radiatif ayant un effet de régularisation instantané. Pour terminer, on propose une approximation de cette équation de transport en terme de Laplacien fractionnaire. Les démonstrations de ces résultats utilisent une analyse asymptotique à partir de la méthode de la fonction test perturbée, des techniques de martingale, ainsi que des représentations probabilistes

In this paper, we study the loss of coherence of a wave propagating according to the Schrödinger equation with a time-dependent random potential. The random potential is assumed to have slowly decaying correlations. The main tool to analyze the decoherence phenomena is a properly rescaled Wigner transform of the solution of the random Schrödinger equation. We exhibit anomalous wave decoherence effects at different propagation scales.