Nonlinear scattering of crossed ultrasonic pulsed beams in the presence of turbulence: Doppler shift and spectral broadening effects

Abstract

The nonlinear scattering of two finite-amplitude mutually perpendicular crossed beams—interacting in the presence of turbulence—generates a sum frequency component that radiates outside the interaction region [Korman and Beyer, J. Acoust. Soc. Am. 84, 339–349 (1988); 85, 611–620 (1989)]. Theoretical results are now extended to predict the angular dependence of Doppler shift and spectral broadening when the incident primary wave components are plane wave pulses instead of continuous waves. Lighthill’s equation is used to find the scattered density ρs(x,t) at the sum frequency. Modeling isotropic turbulence that is stationary, homogeneous, Gaussian, and ‘‘locally frozen,’’ one obtains the ensembled average Fourier spectrum 〈|ρs(x,ω)|2〉=B∫ ∫dω′ dω″|a1(ω′)|2×|a2(ω″)|2W(|ω|−ω′−ω″)ninjΦij(K), where B=(ω2ρ01ρ02m⋅n/c3ρ0x)2(2πcS/|m|), ρ01, ρ02, and ρ0 represent amplitudes and ambient density, n01, n02, and n are incident and scattered unit vectors, m=n01+n02, S=unit area, a1,2(ω)=Fourier amplitude spectrum for each primary pulse, K=(|ω|n−ω′n01−ω″n02)/c, W(ω)=(1/2πKσ)×exp[−(ω−K⋅U)2/2K2σ2], Φij(K) is the turbulence energy spectrum tensor and U, σ, and c are the mean, rms turbulent velocity, and sound speed, respectively.