High-Frequency Wave Propagation in Random Media—a Unified Approach

Abstract

High-frequency wave propagation in a random medium over long propagation distances, where the fluctuations in the wave field are not small and caustic formation is likely, is studied. Techniques based on the geometrical optics approximation or ray theory as well as those based on the parabolic wave equation are examined. It is shown that for small fluctuations in the refractive index of order $\sigma ( {\sigma \ll 1} )$ and long propagation distances of order $\sigma ^{ - 2/3} $, that both these methods are equivalent at high frequencies, at least away from caustics. Since previous theories have identified this $\sigma ( \sigma ^{ - 2/3} )$ distance scale as that on which random caustics first appear, we also derive a regularization of the ray approximation, which is called the beam method, and which is uniformly valid in the vicinity of caustics. On the $\sigma ^{ - 2/3} $ scale, the random ray process converges to a diffusion Markov process, and equations can be derived for the joint distribution of an arbitrary number of ray positions and ray tube areas. It is thus possible to compute arbitrary moments of the wave field for both the rays and the beams. It is shown that the parabolic wave equation can also be scaled for a Markovian limit for propagation distances of order $\sigma ^{ - 2/ 3} $ from which we derive the parabolic moment equations. This is the first derivation of the moment equations that gives the distance scale on which they are valid. The first and second moments can be computed explicitly for both the rays and beams. The results agree with the parabolic equation. The fourth moment of the field at four distant points as computed using ray theory is seen to be singular when all the points coalesce; i.e., the computed scintillation index diverges. The correct result is obtained using the beam method and yields \[ \left\langle {| v |^4 } \right\rangle = \log ( \omega \sigma ^{2/3} )f( t ) + O( 1 ). \] Here t is the scaled propagation distance and $\omega $ is the frequency. The function $f( t )$ is universal depending on the statistics of the medium only through a single distance scale parameter and can be related to the probability of caustic formation.