Stochastic theory of waves and localization in two-dimensional random medium

Random wave field satisfying two-dimensional (2D) wave equation with a random medium is studied using the stochastic representation in the polar coordinate system. The random medium is assumed to be a homogeneous and isotropic Gaussian random field, and its polar spectral representation is effectively employed in the method of solution. The isotropy of a random field implies the invariance under stochastic rotational transformation Dφ, the operator operating on both angular coordinate and probability parameter, and a random wave field with an eigenvalue eiMφ under Dφ, M being an integer, represents an angular mode of random cylindrical wave, and M = 0 corresponding to an isotropic mode; the situation being analogous to the case of cylindrical mode in 2D free space. For a concrete approximate solution, the random medium is assumed to be an omnidirectional random grating with an isotropic narrowband spectrum centered at 2k twice as much as the wave number k, which creates the Bragg scattering in the propagation of random cylindrical waves. The asymptotic form of the random cylindrical wave is obtained by means of the stochastic spectral representation of the random wave, as well as the random medium, and is shown to have an exponential factor exp[γkr]/√kr, γ = ±α + iβ, where the average exponent coefficient γ is given in terms of the spectral density of random medium. This exponential behavior due to the index α in the amplitude describes the localized nature of an angular mode of random wave field generated by a source or by resonance. In the case of a narrowband grating with the variance σ2 and the spectral bandwidth kΔ, γ is concretely calculated to be proportional to σ2 / √Δ, which should be compared with the one-dimensional (1D) case where the corresponding exponential coefficient is proportional to the spectral height σ2/Δ.