High-frequency coherence functions propagated along ray paths in the inhomogeneous background of a weakly random medium: I—Formulation and evaluation of the second moment

Abstract

Ray trajectories have previously been shown to play an important role in determining the transport properties of high-frequency coherence functions constructed from parabolically approximated wave fields. In a weakly fluctuating medium with inhomogeneous refracting background, the parabolic approximation has conventionally been made around a straight propagation direction, thereby restricting its validity to a narrow angular domain with respect to this direction. This limitation is now removed by implementing the parabolic approximation in local coordinates around the curved ray paths connecting a source with an arbitrarily located observer in the background medium. The corresponding partial differential equation for the two-point coherence function is solved by a multiscale expansion method. Solution strategies are distinguished according to whether the observer is reached by a single ray, by two or more rays with vastly differing trajectories, or by two or more closely adjacent rays. In the first instance and also in the second (because the fluctuations decorrelate the separate ray fields), the coherence function can be synthesized by contributions from each individual path. In the third instance, which represents observations near caustics or foci, the adjacent ray fields must be treated collectively. The resulting integral expressions for the intensities can be simplified when the excitation is due to a line source and fluctuations are omitted. The reduced explicit forms are found to agree with the ray acoustically computed deterministic intensities at regular observation points and near simple caustics, thereby lending confidence in the validity of our approach and the quality of our approximations. These asymptotic ray acoustic studies of coherence functions have also been extended to phenomena involving reflection, refraction, and diffraction at interfaces and boundaries.