Parabolic-Approximation Models for Acoustic and Electromagnetic Wave Propagation

A new parabolic equation (PE) is presented that is independent of k_0 and capable of handling relatively large range variations in the index of refraction. This equation is similar to, and ostensibly simpler than, an earlier range refraction PE (RAREPE). The modified range reaction parabolic equation (MOREPE) is obtained by a transformation approach, and operator and multiscale formalisms are described to validate the equation. Principal properties of MOREPE are developed, including energy conservation and possession of the correct (Helmholtz) rays in the high -frequency, small-angle limit. Exact solutions with range variation in sound speed are presented to illustrate differences between standard PE (SPE) and MOREPE. Propagation examples in range-independent environments demonstrate close agreement between MOREPE and SPE, while examples with strong range dependence exhibit significant differences between the two equations in their predictions of acoustic intensity. Analytical and numerical comparisons of solutions to the one-way Helmholtz equation (HE1), MOREPE, and SPE demonstrate the increased accuracy of MOREPE over SPE in range-dependent environments. A PE method for the prediction of coherent, low -frequency acoustic propagation through small-scale atmospheric turbulence is presented. Frequency constraints on the applicability of stochastic parabolic approximations are avoided by first averaging the stochastic Helmholtz equation and then applying a parabolic approximation to the resulting deterministic equation. Turbulence effects are incorporated by means of spatially-varying effective wave numbers. Comparison of exact solutions in the case of infinite-space propagation demonstrates the advantages and limitations of this approach. A uniform asymptotic expression for the effective wave-number profile in the case of isotropic turbulence is used to develop a half-space PE formulation that is valid in the limit of low frequency, small-scale inhomogeneity. For anisotropic turbulence that is correlated more strongly in range than height, a modified mean-value theorem for the 2-D Helmholtz operator is used to find the effective wave number. Numerical examples demonstrate that excess attenuation due to the imaginary component of the effective wave number is the primary effect of weak turbulence on coherent, low-frequency propagation. Parabolic approximation models for anisotropic electromagnetic wave propagation in the atmosphere are developed. Particular attention is paid to the lower end of the high-frequency band where the effects of atmospheric inhomogeneity and anisotropy due to magnetic coupling are most prominent. Derivations of model equations for the electric field are based upon transformation and operator formalisms employed previously by the authors in an analogous scalar problem. These formalisms are designed to avoid parameter-size restrictions of existing parabolic-approximation methods. A priori knowledge of electric-field polarization is not required. Reduction of the model to a model developed recently by Brent et al. is described in an appropriate limiting case.