Wave Propagation in Anisotropic Media Using Born Series Through 4th Order

Abstract

The Born series is used to calculate a field propagating in a Gaussian medium which itself has a Gaussian-form correlation function. These specializations lead to integrals for the moments with integrands which are themselves mainly of Gaussian form. They can be integrated algebraically using multivariate Gaussian distribution formulas. Specifically for moments the, say, fourth order Born term involves a 12-fold integration of which eight can be done algebraically using the formulas. We are able, for a two-scale anisotropic medium, to sum the Born series for the average (the first moment) of the radiation field. The result reduce to familiar ones when we specialize to isotropic media. A novel characteristic is that for physically interesting parameter ranges diffraction effects can be substantially reduce the phase variance, thus reducing the randomizing effect of the medium on the propagated field. Anisotropy also leads to an intensity dependance on angle. The fourth-order moment (the intensity variance) is considered through fourth-order Born terms; it is not possible to sum the Born series for this moment in terms of simple functions even for isotropic media. Previous results are in large part confirmed: when diffraction effects are slight, sound field fluctuation effects are confined to the phase, leading at sufficient range to a log-normal field distribution. When diffraction effects become appreciable, the sound amplitude fluctuates and the distribution becomes Gaussian. There is also an expected region of appreciable overshoot in the intensity variance, that variance exceeding its Gaussian value for intermediate ranges at lower frequencies.