spectrum of the amplitude fluctuations depends 
strongly on the decay rate of the thermal vari- 
ation spectrum over the small components, or 
high space wave numbers. 
2. When the smallest component of the thermal 
variations is less thanVAL ( as is generally 
the case in sonar ), then the components whose 
dimensions are of orderVAL have the greatest 
effect on.the spectrum of the amplitude fluctu- 
ations. As an example, at a range of 4000 yards 
at 20 kc. the thermal variations of dimensions 
of the order of 18 yards have the greatest 
effect on the amplitude fluctuations. This 
implies that in order to gather the statistics 
of the thermal variations which most affect the 
amplitude fluctuations the temperatures must 
be sampled over distances less than 18 yards. 
3. The spectrum of the phase fluctuations is 
principally affected by the large size components 
of the thermal variations, while the mean square 
fluctuation of the wave front normal direction 
is principally affected by the large to middle 
Size components. 
4, Should the form of the spectrum of the 
thermal variations differ considerably from 
the one used in the analysis, represented 
by the Von Karman interpolation formla, the 
conclusions reached above may differ consider- 
ably, especially for the amplitude fluctuations 
which are most sensitive to the form of the 
thermal variation spectrum. Thus, measurements 
ere needed to establish the form of the thermal 
variation spectrum, which requires making 
Measurements over small as well as large 
distances. 
5. In order to overcome the effects of phase 
and amplitude fluctuations, statistical pro- 
cessing of sonar signals seems in order. If 
the random thermal structure changes rapidly 
in time then a good statistical sample of 
target bearings can be obtained which when 
processed will reduce the errors due to 
the fluctuations. This implies that all the 
component sizes in the thermal variations 
change rapidly in time. It is more likely, 
however, that the larger components will 
vary more slowly in time than the smaller 
components, Under this condition, it becomes 
necessary that the statistics of the thermal 
variations in time also be investigated in 
order to establish the effectiveness of 
statistically processing the fluctuating 
sonar data. 
REFERENCES. 
1. Tatarski, V.I., "Wave Propagation In A 
Turbulent Medium", McGraw-Hill Book Co.,Inc., 
New York, 1961. 
2, Hinze, J.0., “Turbulence, An Introduction 
to Its Mechanism and Theory", McGraw-Hill 
Book Co., Inc., New York, 1959, Chapter 3. 
38 
