TR No. 22 
For a change in x’ of 100 meters (the average sample length) the 
change in variance is about 5,3 cm?=sec™“, which is not significant 
compared to the statistical variations among successive samples, The 
large variations are attributed to inhomogeneity of the field of turbulence, 
short sample lengths, and non-linear variations in the towing velocity. 
A more precise indication of the accuracy of the results is obtained 
from the energy spectra. A measure of the accuracy of any computed value 
of the energy spectrum is the equivalent number of degrees of freedom of 
the value (Blackman and Tukey, 17). The equivalent number of degrees of 
freedom is approximately given by 
2(sampie length) 
maximum lag 
je 8 
which for all of the samples is 
k = 2(500) = 20 degrees of freedom. 
50 
The distribution of computed values of the energy spectrun@,,A/opta ined 
from a large number of similar samples having an equivalent number of degrees 
of freedom, k, is assumed to be equal to a Chi-Square distribution with 
k degrees of freedom, That is 
8 GOS) Bee 
Uk) (34) 
where U(X) is the value of the energy spectrum function that would be obtained 
from a sample of infinite length. Using this assumption, confidence limits 
can be assigned to the computed values of the energy spectrum function. From 
the tables in reference 18 values of X* corresponding to the probabilities of 
occurrence of deviations greater than Yrcan be found, For a probability of 
0.10 of a deviation greater than Lae the value of £* for 20 degrees of freedom 
is 28.412, Similarly, for a probability of 0.90 Z*= 12,443, Thus the prob- 
ability is 0.80 that the deviation from Z’is within the interval 12.443 to 
28.412, or that 
KP, U0 
Lk ) 
23 
Nyars IG 222, 4 12 
