210 Lecture 12 
of the measurement, and is considerably greater than the intensity predicted 
by the Kolmogorov equilibrium law (see Fig. 12.6). However, as the patch 
size becomes smaller, most of the fluctuations caused by the weather and 
by the daily variations of the temperature seem to average out, and a stable 
Kolmogorov-type space spectrum is generated. The slope of the curves in 
Fig. 12.5, then, is exactly one third. When the space wavelength becomes very 
small, the effects of viscosity and heat conduction predominate, and the tem- 
perature fluctuation eventually decreases inversely to the seventh power of the 
space wavelength [4]. This range does not appear in Fig. 12.5. 
The space spectrum of the temperature fluctuations of the sea can be divided 
into three different regions; (1) the long wave number range, (2) the equilibrium 
Kolmogorov range, and (3) the dissipation range. The long wave number range, 
which depends greatly on the depth of the measurement, and which also seems 
to depend greatly on the weather and its history, as will be discussed in the next 
section, has practically no effect on sound scattering or the fluctuations of the 
transmitted signal. These are predominantly determined by the spectral com- 
ponents of the temperature pattern that have a wavelength A,, equal to, or greater 
than, half the sound wavelength up to wavelengths somewhat above that given 
by r=kR’, where r is the range, and R ~X,,/4 represents the radii of the tem- 
perature patches. For the sound frequencies of practical interest, the spectral 
components of the temperature structure that affect sound propagation are 
always within the Kolmogorov equilibrium range. 
The spectral distribution curve of the temperature fluctuations in water, as 
far as they affect sound propagation, can be assumed to be independent of the 
weather or the external conditions of the Kolmogorov type, E(k)= Kx- 3, This 
assumption leads to a considerable simplification in the theories of sound scat- 
tering and of the fluctuations of the transmitted signal. All that needs to be de- 
termined in the Kolmogorov law is the constant K as a function of the depth and 
of the variations in the weather. This can be done by measuring the rms 
temperature fluctuation between two points at a constant distance from one 
another. The results obtained in the past by acoustic measurements seem to 
indicate that the value of this constant decreases with depth, but depends only 
slightly on the weather. 
The constant K can be expressed as a function of the mean-square deviation 
of the sound velocity. If the power spectrum is constant up to a space wave 
number x), and if it obeys a simple power law for the higher space wave 
numbers, 
E(k)=Kx-™ KS Ko 
(12) 
E(k)=Kko™ K2 Ko 
the mean-square value of the velocity fluctuations then becomes 
KO © ha : 
a? =f E(x) de = Kea" + [Kee dk =—@ Kd (13) 
The constant K, therefore, is given by 
K=Ma1 po q? (14) 
