Temporal frequency spectra of multifrequency waves in turbulent atmosphere

General formulations for temporal frequency spectra of the fluctuations of plane, spherical, and beam waves operating at two frequencies are given based on weak turbulence and frozen-in assumptions. The cross spectra and the coherence are obtained for the amplitude at two frequencies, the phase at two frequencies, and the amplitude at one frequency and the phase at another frequency. The results are examined in detail for plane and spherical waves. For the spectrum of the index of refraction
\kappa^{-n}
in the inertial subrange, the amplitude spectrum behaves as
k^{(5-n)/2}
for
\omega \rightarrow 0
and
k^{2}\omega^{1-n}
for
\omega \rightarrow \infty
. The phase spectrum for
\omega\rightarrow 0
and for
\omega\rightarrow\infty
behaves as
k^{2}\omega^{1-n}
with different constants. These results agree well with the experimental work of Janes et al. [11] at 9.6 and 34.5 GHz, and explains the ratio of the spectra at two frequencies. Also noted is the experimental slope of -2.6 as and for
\omega \rightarrow \infty
which may be compared with
1-n = -2.66
using the Kolmogorov spectrum of
n = 11/3
. The amplitude and phase coherence are calculated, and the results agree well with the experimental data. This agreement is indicative of the general validity of the theory for frequencies as low as
10\sim30
GHz and the path length as long as 60 km. It is also shown that using the preceding theory, the wind velocity and the structure constant
C_{n}
can be deduced from the experimental data. Theoretical wind velocity of 15.6 knots obtained from the propagation data compares favorably with the meteorologically measured value of 14 knots, and two values of
C_{n}
obtained independently from the amplitude and phase measurements closely agree with each other.