Spectral broadening of waves propagating in a random medium

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Abstract
Formulas for the cross-correlation and spectral density functions of the scalar wave field radiated by a random point source in a weakly inhomogeneous three-dimensional time-dependent random medium are derived. The medium is assumed to be statistically homogeneous and isotropic and to be statistically independent of the source. The analysis is based on a modification of the smoothing method. An approximate expression for the power spectrum of the wave as a function of the source-field point distance (or propagation distance) is obtained for the case in which the characteristic frequency of the source is much greater than that of the medium. This expression shows that the wave spectrum approaches a limiting form, which is referred to here as the fully developed spectrum, with increasing propagation distance. It is also found that the total signal power is conserved as the spectrum evolves. Results obtained for the case of a narrow-band source indicate that the spectral bandwidth increases initially as the square root of the propagation distance, but that at larger distances it approaches a limiting value. Numerical results obtained for the narrow-band case show a progressive broadening of the wave spectrum with increasing propagation distance and/or with increasing strength of the randomness of the medium, in agreement with observations.