Directed waves in random media: An analytical calculation

Journal Article · Wed Jun 01 04:00:00 EDT 1994 · Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics; (United States)

DOI:https://doi.org/10.1103/PhysRevE.49.5755· OSTI ID:7112688

Friedberg, R ^[1]; Yu, Y ^[2]

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1. Physics Department, Barnard College and Columbia University, New York, New York 10027 (United States)
2. Physics Department, Columbia University, New York, New York 10027 (United States)

The propagation of directed scalar waves in [ital D]+1 dimensions in a strongly disordered medium is studied. We use the model first proposed by Saul, Kardar, and Read [Phys. Rev. A 45, 8859 (1992)], where unitarity is guaranteed in each step. The beam positions [l angle][bold x][sup [bar 2]][r angle] and [l angle][bar x][sup 2][r angle] characterize the transverse fluctuations of a directed wave front, where the overbar means an average over the wave profile for a given realization of randomness, and [l angle] [r angle] means a quenched average over all realizations. We introduce [ital G][sub [ital q]][sup [bold k]]([bold y]) as the Laplace-transformed Green function of two free random walkers with center-of-mass momentum [bold k] and relative position [bold y]. We calculate analytically the mean-square deviation of the beam center, [l angle][bar x] [sup 2][r angle], as a function of time. The results show that, for large [ital t], [l angle][bar x][sup 2][r angle] behaves as (1/ [radical][pi] )[ital t][sup 1/2][minus]1/4+[ital O]([ital t][sup [minus]3/2]) in 1+1 dimensions and as (ln[ital t]+4 ln2+[gamma])/4[pi]+[ital O](1/[ital t]) in 2+1 dimensions and takes the finite value 1/2[ital D][[ital G][sub [ital q]=1][sup [bold k]=0](0)[minus] [radical](27/4[pi]) t[sup [minus]1/2][delta][sub [ital D],3]]+[ital O](1/[ital t]) in [ital D]+1 dimensions where [ital D][ge]3, [gamma] being the Euler constant. We generalize these results to a twofold random walk with any probability-flux-conserving interaction. In all cases the leading term at large [ital t] depends solely on the finite value or leading singularity of [ital G][sub [ital q]][sup [bold k]=0](0) at [ital q]=1.

• OSTI ID: 7112688
• Journal Information: Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics; (United States), Journal Name: Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics; (United States) Vol. 49:6; ISSN 1063-651X; ISSN PLEEE8
• Country of Publication: United States
• Language: English

Related Subjects

661100* -- Classical & Quantum Mechanics-- (1992-)
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ANALYTICAL SOLUTION
BEAM DYNAMICS
FLUCTUATIONS
FUNCTIONS
GREEN FUNCTION
RANDOMNESS
REFRACTION
VARIATIONS
WAVE PROPAGATION