Theory of the backscattering of sound by phase-matched nonlinear interaction

Abstract

The nonlinear interaction of noncollinear acoustical waves is considered. Conditions of the resonant backscattering of one wave from the lattice produced by the other two are formulated in analogy with the four-wave mixing known in optics. The efficiency of the phase-matched interaction of acoustical waves is calculated in the resonant approximation for a gas media. Such approximation is constructed on the basis of the expansion of the sound equations preserving up to cubic terms. The amplitude of the backscattered wave is expressed as the product of the efficiency, the amplitudes of three waves, the wave number of the backscattered wave, and the size of the region of interaction. Such backscattering is proposed as an acoustical remote probe. The distance to the interaction region and the amplitude of initial waves are limited by nonlinear degradation of waves due to the second-order nonlinearity. For acoustical waves with wave number 10 m−1, sources of size 1 m, and about 100 m to the interaction region, the amplitude of the backscattered wave can be about 10−10 of the atmospheric pressure. At the detection with a signal-to-noise ratio about of 10, the resolution of such method on the wind velocity may be about 1 m/s.