EFFECTS OF THE SPECTRAL COMPOSITION OF RANDOM 
THERMAL VARIATIONS ON PHASE AND AMPLITUDE 
FLUCTUATIONS OF A SOUND WAVE PROPAGATING IN THE SEA 
by AIMO SALENIUS 
Sperry Gyroscope Company 
Great Neck, New York 
ABSTRACT. 
Equations are derived for the spectral densities of 
the phase and amplitude fluctuations, as well as the mean 
square fluctuation of the wave front normal direction in 
terms of the spectral density of the index of refraction 
variations for the wave of a harmonic point source in a 
slightly random medium whose mean index of refraction 
is constant. Since in the sea the spectral density of the 
refractive index fluctuations is, for all practical purposes, 
determined as a multiple of the spectral density of the 
thermal variations, the equations permit the study of the 
effects of the spectral composition of the random thermal 
variations over various space wavelengths in order to 
determine the sampling distances required to obtain rel- 
evant statistics of the thermal variations. 
I. INTRODUCTION. 
The phase and amplitude fluctuations of a sound 
wave propagating in the sea have considerable effects 
on reducing the quality of performance of sonars, The 
phase fluctuations contribute to bearing errors while the 
amplitude fluctuations create difficulties in detection and 
signal processing. It is well known that a major portion 
of these fluctuations are caused by the random thermal 
structure of the sea which causes the speed of sound to 
vary randomly from point to point in the medium. 
In the following analysis we shall assume that the 
mean temperature is known throughout the medium and 
that the fluctuations about the mean are stationary to the 
second order. Furthermore, in order to make the problem 
tractable we shall also assume the random field of the 
temperature variations to be homogeneous and isotropic. 
A better characterization of the temperature variations 
would be to consider them axisymmetric about the depth 
axis, however, the additional complexities would not 
Significantly contribute to the qualitative aspects of the 
results, 
One may consider the thermal variations as con- 
sisting of a continuous distribution of periodic variations 
of random amplitudes over various space wavelengths, or 
component sizes. The average power of the fluctuations 
can be represented by means of a continuous distribution 
of power density over the various size components. This 
is achieved mathematically by the Fourier transform of 
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the autocorrelation function of the temperature variations. 
The resulting power density spectrum will be simply re- 
ferred to as the spectrum of the thermal variations. For 
small variations of temperature, neglecting the effects of 
the extremely small variations in salinity, the power den- 
sity spectrum of the index of refraction variations is de- 
termined as a multiple of the power density spectrum of 
the thermal variations. Thus, when we refer to the spec- 
trum of the index of refraction variations we are also re- 
ferring to the spectrum of the thermal variations. 
We shall be concerned particularly with the effects 
on the phase and amplitude fluctuations of the manner in 
which the power of the thermal variations is distributed 
among its various size components. Knowing the effects 
of the thermal fluctuations of various size components, 
it will be possible to plan experiments for gathering the 
statistics of the temperature variations such that they 
will be useful in the study of sonar performance. In par- 
ticular, it will be possible to determine the smallest sam- 
pling distances required to secure adequate statistics. 
Since target bearings obtained by correlation sonars 
are determined by measuring the orientation of the wave- 
front, we shall also investigate the effects of the spectrum 
of the thermal variations on the mean square fluctuation 
of the direction of the normal to the wave front in order 
to get a measure of sonar bearing accuracies. 
In most sonar applications the sound source is 
considered as a point source and the sound field is very 
nearly spherical. For this reason we shall study the case 
of a harmonic point source in a medium whose mean index 
of refraction is a constant. Furthermore, since correla- 
tion sonars are concerned primarily with the correlation 
of signals along a base line transverse to the direction of 
wave propagation, we shall concern ourselves with the 
spectra of the transverse phase and amplitude fluctuations. 
Tatarski’ has analyzed the problem of the mean 
square fluctuation of phase and amplitude at a point in 
the field of a point source in a random medium whose 
mean Structure is constant. The results herein are ob- 
tained by a modification and extension of the approach 
used by Tatarski in order to determine as well the trans- 
verse phase and amplitude spectra, and the mean square 
fluctuation of the wave front normal direction. 
