Breaker and O'Neil 



required for the pattern of the turbulence to shift appreciably and 

 if the decibel level (with the average power as a reference) is 

 observed, in a plane orthogonal to the x axis, the mean square level 

 is found to be. 



X^= 197.10 C^ X^« l"/" , vOTl »4 (3) 



where X is the acoustic wave length and L is the range. Moreover, 

 one can find the two dimensional spectrum of the level fluctuations 

 in the plane as* 



2 ..2 . /' . K . K^L 



¥^ (/c,0) =2.48967rCn K L I j^-sin 



K 



K 



■1/3 



L K / (4) 



where k is the wave number of the acoustic wave (k=27r/X), /Ct is the 

 wave number of the fluctuation in the y direction, /c, that in the z 

 direction, and, in consequence of the isotropy assumption, /c^ = Kg -^ k\- 



The case of practical interest is that in which one is 

 moving along some line in the plane x = constant with velocity v^. 

 This could represent either actual movement of the receiver with 

 respect to the source or displacement of the water (e.g., by a 

 current with velocity Vj^) across the x axis. (Velocity components 

 along the x axis, unless very large, will have little effect upon 

 the fluctuations.) If the turbulence structure may be regarded as 

 "frozen-in" the fluid (see below), then the total variance observed 

 will still be represented by Eq. (4) but the (one-dimensional) 

 spectrum of the fluctuations in level will be 



8 TT / ^ M 2 . 4 TT f 



'^K-f^ 



^ ^n ' 

 where f is the frequency of fluctuation. This may be reduced to the 



dimensionless quantity. 



o,n-lMiI , ^^ fl sin(t'+u^) 1 ,.2. 2:'^6, . 



P(r) = 2— = l.35u/ I 2 — 2 (t+u) dt 



o" y L t +u J 



(S) 



*To aid in interpreting F.(/c ,0) , recall that the spectrum P(f) of an 

 ordinary one-dimensional time-random variable, x(t), records the 

 contribution made to the variance (or mean-square fluctuation) of x 

 by frequencies near f . Since the wave numbers, /c- and /c, may be 

 interpreted as spatial (angular) frequencies F^ simply gives the 

 contribution made to the variance of the fluctuation by spatial 

 frequencies lying in a small "area" near K 2 and k ^. Isotropy 

 allows one to pick any k o and k^ lying on a circle k^* k -i ~ 

 constant and get the same result, so F^ can be formally expressed 

 as a function of one variable, k . 



369 



