Theory of transfer with delay for trapping of nonstationary acoustic radiation in a resonant randomly inhomogeneous medium

The propagation of a quasimonochromatic wave packet of acoustic radiation in a discrete randomly-inhomogeneous medium under the condition that the carrier frequency of the packet is close to the resonance frequency of Mie scattering by an isolated scatterer is studied. The two-frequency Bethe-Salpeter equation in the form of an exact kinetic equation that takes account of the accumulation of the acoustic energy of the radiation inside the scatterers is taken as the initial equation. This kinetic equation is simplified by using the model of resonant point scatterers, the approximation of low scatterer density, and the Fraunhofer approximation in the theory of multiple scattering of waves. This leads to a new transport equation for nonstationary radiation with three Lorentzian delay kernels. In contrast to the well-known Sobolev radiative transfer equation with one Lorentzian delay kernel, the new transfer equation takes account of the accumulation of radiation energy inside the scatterers and is consistent with the Poynting theorem for nonstationary acoustic radiation. The transfer equation obtained with three Lorentzian delay kernels is used to study the Compton-Milne effect—trapping of a pulse of acoustic radiation diffusely reflected from a semi-infinite resonant randomly-inhomogeneous medium, when the pulse can spend most of its propagation time in the medium being “trapped” inside the scatterers. This specific albedo problem for the transfer equation obtained is solved by applying a generalized nonstationary invariance principle. As a result, the function describing the scattering of a diffusely reflected pulse can be expressed in terms of a generalized nonstationary Chandrasekhar H-function, satisfying a nonlinear integral equation. Simple analytical asymptotic expressions are found for the scattering function for the leading and trailing edges of a diffusely reflected δ-pulse as functions of time, the reflection angle, the mean scattering time of the radiation, the elementary delay time, and the parameter describing the accumulation of radiation energy inside the scatterers. These asymptotic expressions demonstrate quantitatively the retardation of the growth of the leading edge and the retardation of the decay of the trailing edge of a diffusely reflected δ-pulse when the conventional radiative transfer regime goes over to a regime of radiation trapping in a resonant randomly-inhomogeneous medium. Zh. éksp. Teor. Fiz. 113, 432–444 (February 1998)