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Coupled paraxial wave equations in random media in the white-noise regime. (English) Zbl 1175.60066
The authors study transmission and reflection of acoustic waves in a slab random medium consisting of 3-dimensional random fluctuations and whose end can be either transparent or a strong interface. Indeed, the authors motivate the use of rapid fluctuations and they prove that the paraxial and white-noise approximations are valid; the convergence to the white-noise paraxial wave equation in the case of a stratified media with weak fluctuations has been studied by F. Bailly, J. F. Clouet and J. P. Fouque [SIAM J. Appl. Math. 56, 1445–1470 (1996; Zbl 0859.60061)]. The limit random model that has been obtained corresponds to a system of coupled Schrödinger equations driven by a Brownian field. Then, owing to this model, the two-point statistics of the transmitted and reflected waves can be computed and the enhanced backscattering phenomenon can be studied.
More precisely, as described in Section 2 of the paper, the linear acoustic waves propagate according to the following governing equations: $\rho(z,x)\frac{\partial u}{\partial t} +\nabla p=F, \quad \frac{1}{K(z,x)}\frac{\partial p}{\partial t} + \nabla u=0,$ where $$p$$ is the pressure field, $$u$$ the velocity field, $$\rho$$ is the density of the medium, $$K$$ is the bulk modulus of the medium, which models the medium fluctuations, and $$(z,x)\in \mathbb{R}\times \mathbb{R}^d$$ are the space coordinates. $$F$$ corresponds to the forcing term. The random slab under consideration is determined by $$z\in (0,L)$$, where $$L$$ is the propagation distance, and it is assumed to lie in-between two homogeneous half-spaces. The main features of the random media are described (in terms of $$K$$ and $$\rho$$) according to the assumption that the medium fluctuations vary rapidly in the random slab $$(0,L)$$. Then, the scaling regime considered in the paper is described in detail so that the transmitted and reflected wave fields are obtained.
Section 3 is devoted to prove that the above mentioned transmitted and reflected wave fields in the scaling regime converges in law, in some space of functions, to a limit which is described by a random Schrödinger model. The main points in the proof are the following: tightness and a priori estimates for the sequence, convergence of the finite-dimensional distributions and use of Itô’s formula for Hilbert-space-valued processes.
Eventually, the two-point statistics of the transmitted and reflected wave fields are computed and the enhanced backscattering property is established. This has been done thanks to a detailed description of the Wigner distributions associated to the transmitted and reflected waves and its corresponding integral representations (Sections 4 and 5).

##### MSC:
 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 35R60 PDEs with randomness, stochastic partial differential equations 74J20 Wave scattering in solid mechanics 60K37 Processes in random environments
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