Coupled paraxial wave equations in random media in the white-noise regime.

*(English)*Zbl 1175.60066The 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).

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).

Reviewer: Lluís Quer-Sardanyons (Bellaterra)

##### 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 |

PDF
BibTeX
XML
Cite

\textit{J. Garnier} and \textit{K. Sølna}, Ann. Appl. Probab. 19, No. 1, 318--346 (2009; Zbl 1175.60066)

**OpenURL**

##### References:

[1] | Bailly, F., Clouet, J. F. and Fouque, J. P. (1996). Parabolic and Gaussian white noise approximation for wave propagation in random media. SIAM J. Appl. Math. 56 1445-1470. JSTOR: · Zbl 0859.60061 |

[2] | Bamberger, A., Engquist, B., Halpern, L. and Joly, P. (1988). Parabolic wave equation approximations in heterogenous media. SIAM J. Appl. Math. 48 99-128. JSTOR: · Zbl 0654.35055 |

[3] | Barabanenkov, Y. N. (1973). Wave corrections for the transfer equation for backward scattering. Izv. Vyssh. Uchebn. Zaved. Radiofiz. 16 88-96. |

[4] | Bernardi, C. and Pelissier, M.-C. (1994). Spectral approximation of a Schrödinger type equation. Math. Models Methods Appl. Sci. 4 49-88. · Zbl 0791.35026 |

[5] | Blomgren, P., Papanicolaou, G. and Zhao, H. (2002). Super-resolution in time-reversal acoustics. J. Acoust. Soc. Am. 111 230-248. |

[6] | Claerbout, J. F. (1985). Imaging the Earth’s Interior . Blackwell Science, Palo Alto, CA. |

[7] | Clouet, J.-F. and Fouque, J.-P. (1994). Spreading of a pulse travelling in random media. Ann. Appl. Probab. 4 1083-1097. · Zbl 0814.73018 |

[8] | Dawson, D. A. and Papanicolaou, G. C. (1984). A random wave process. Appl. Math. Optim. 12 97-114. · Zbl 0564.60061 |

[9] | Fannjiang, A. C. (2005). White-noise and geometrical optics limits of Wigner-Moyal equation for beam waves in turbulent media. II. Two-frequency formulation. J. Stat. Phys. 120 543-586. · Zbl 1081.78004 |

[10] | Fannjiang, A. C. (2006). Self-averaging radiative transfer for parabolic waves. C. R. Math. Acad. Sci. Paris 342 109-114. · Zbl 1088.35054 |

[11] | Fannjiang, A. and Sølna, K. (2005). Superresolution and duality for time-reversal of waves in random media. Phys. Lett. A 352 22-29. |

[12] | Fannjiang, A. C. and Solna, K. (2005). Propagation and time reversal of wave beams in atmospheric turbulence. Multiscale Model. Simul. 3 522-558 (electronic). · Zbl 1075.35084 |

[13] | Feizulin, Z. I. and Kravtsov, Yu. A. (1967). Broadening of a laser beam in a turbulent medium. Radio Quantum Electron. 10 33-35. |

[14] | Fouque, J.-P., Garnier, J., Papanicolaou, G. and Sølna, K. (2007). Wave Propagation and Time Reversal in Randomly Layered Media. Stochastic Modelling and Applied Probability 56 . Springer, New York. · Zbl 1386.74001 |

[15] | Fouque, J. P., Papanicolaou, G. and Samuelides, Y. (1998). Forward and Markov approximation: The strong-intensity-fluctuations regime revisited. Waves Random Media 8 303-314. · Zbl 0915.35119 |

[16] | Garnier, J., Gouédard, C. and Videau, L. (2000). Propagation of a partially coherent beam under the interaction of small and large scales. Opt. Commun. 176 281-297. |

[17] | Garnier, J. and Sølna, K. (2008). Effective transport equations and enhanced backscattering in random waveguides. SIAM J. Appl. Math. 68 1574-1599. · Zbl 1152.76042 |

[18] | Garnier, J. and Sølna, K. (2008). Random backscattering in the parabolic scaling. J. Stat. Phys. 131 445-486. · Zbl 1146.78006 |

[19] | Ishimaru, A. (1997). Wave Propagation and Scattering in Random Media . IEEE Press, New York. Reprint of the 1978 original, with a foreword by Gary S. Brown, An IEEE/OUP Classic Reissue. · Zbl 0873.65115 |

[20] | Miyahara, Y. (1982). Stochastic Evolution Equations and White Noise Analysis. Carleton Mathematical Lecture Notes 42 1-80. Carleton Univ., Ottawa, Canada. · Zbl 0489.60070 |

[21] | Papanicolaou, G., Ryzhik, L. and Sølna, K. (2004). Statistical stability in time reversal. SIAM J. Appl. Math. 64 1133-1155 (electronic). · Zbl 1065.35058 |

[22] | Ryzhik, L., Papanicolaou, G. and Keller, J. B. (1996). Transport equations for elastic and other waves in random media. Wave Motion 24 327-370. · Zbl 0954.74533 |

[23] | Strohbehn, J. W., ed. (1978). Laser Beam Propagation in the Atmosphere . Springer, Berlin. |

[24] | Tappert, F. D. (1977). The parabolic approximation method. In Wave Propagation and Underwater Acoustics. Lecture Notes in Phys. 70 224-287. Springer, Berlin. |

[25] | Tatarskiĭ, V. I., Ishimaru, A. and Zavorotny, V. U., eds. (1993). Wave Propagation in Random Media ( Scintillation ). Copublished by SPIE-The International Society for Optical Engineering, Bellingham, WA. Papers from the International Meeting held at the University of Washington, Seattle, Washington, August 3-7, 1992. |

[26] | Thrane, L., Yura, H. T. and Andersen, P. E. (2000). Analysis of optical coherence tomography systems based on the extended Huygens-Fresnel principle. J. Opt. Soc. Amer. A 17 484-490. |

[27] | van Rossum, M. C. W. and Nieuwenhuizen, Th. M. (1999). Multiple scattering of classical waves: Microscopy, mesoscopy, and diffusion. Rev. Modern Phys. 71 313-371. |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.