The power spectrum of sound received from a known acoustic source after transmission through a region of turbulence is calculated by a convolution with a distortion function. The latter specifies the modification of the acoustic power spectrum as it propagates through the turbulent region. The distortion function is the Fourier transform with regard to time of the characteristic function of a bivariate aleatory process, whose random variables are the acoustic phase shifts of two neighbouring waves; the phase shifts are due to the Doppler effect of the random turbulence velocity and are a function of position and time. The turbulence spectrum is used to calculate the r.m.s. phase shift and also the correlation time in the correlation coefficient. This specifies the received power spectrum for any type of source, viz a monochromatic or tonal source, or a broadband source, e.g. Gaussian or exponential. The received power spectra depend on the choice of correlation function, e.g. Gaussian or volume conserving. The received power spectra simplify in the asymptotic case of large r.m.s. phase shift; otherwise, the exact received power spectra consists of an attenuated source spectrum plus a series of scattering broadbands. The latter demonstrate how the acoustic energy in the emitted source spectrum is spread over a wider range of frequencies by propagation through turbulence, leading to spectral broadening in the received spectrum.
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