482 



VELOCITY AND TEMPERATURE STRUCTURE 



ture varies rapidly from point to point. For sim- 

 plicity, suppose that positive and negative values of 

 c are equally likely — that is, that the average tem- 

 perature of the water is just equal to the temperature 

 outside the scattering medium. Although the distri- 

 bution of temperature from point to point is a 

 quantity which fluctuates at random, there is a 

 certain patch size over which the temperature does 

 not usually change appreciably. This is represented 

 mathematically by means of the function p{^), which 

 is defined by the expression 



P(f) 



Acix + ^,%j,z) Ac{x,y,z) 



Ac{x,y,zY 



(7) 



where the averaging is to be carried out in space, over 

 all values of x, y, and z in the scattering region. While 

 f is a displacement in the x direction in the expression 

 (7) above, the displacement might also be extended 

 in any other direction. The value of the function p(f ) 

 will depend both on the magnitude and on the direc- 

 tion of f . If the displacement is zero, then p will equal 

 unity. If the displacement is very large, the values of 

 c at points separated by the distance r show no cor- 

 relation with each other, and their product is alter- 

 nately positive and negative, canceling out on the 

 average; thus for large f, p approaches zero. The 

 patch size is the value of f for which p becomes small, 

 say less than about 3^^. The function p is called a self- 

 correlation coefficient. The temperature microstructure 

 is described as isotropic if p(f) is independent of the 

 direction along which f is taken. 



With some mathematical transformations, equa- 

 tion (5) may be expressed in terms of p(f). For an 

 isotropic medium, the resulting equation, which is 

 equivalent to that given in a report by Columbia 

 University Division of War Research [CUDWR],^ 



where V is the volume of the scattering region. As one 

 fairly general type of possible correlation coefficient, 

 it may be assumed 



P(f)=e-^'^. (9) 



By substituting this expression in equation (8), and 

 integrating, 



1 



_fc J_ /AcV 

 V ^ 8wA\c) 



— ■ (10) 



In actual practice the wavelength X is usually less 



than the patch size A, and the last term in the de- 

 nominator may be neglected. Correlation coefficients 

 of a form different from equation (10) do not gener- 

 ally give a much greater value of k/V for a given 

 patch size A. 



Numerical values may be substituted in equation 

 (10). Fluctuations of 0.5 F with a patch size of 6 in. 

 probably represent a rather extreme assumption. For 

 this situation, k/V is about 3 X 10~' sq yd per cubic 

 yard of volume. The volume scattering coefficient m 

 discussed in Section 12. 1 of this volume is related to k 

 by the equation 



m = — ■ (11) 



Thus m, in this case, is about 4 X 10"'' per yard. If 

 equal energy were scattered in all directions, m would 

 be the fraction of energy scattered per yard of sound 

 travel through the scattering medium. 



Evidently even these extreme assumptions give a 

 very small scattering coefficient. Even if the scatter- 

 ing volume is 10 yd thick, 30 yd across, and 100 yd 

 long, corresponding to the wake in the path of a 

 sound beam, k is about 10~' yd, corresponding to an 

 effective target strength of —30 db. The transmission 

 loss through such a scattering region would be a very 

 small fraction of a decibel. Temperature microstruc- 

 ture cannot explain the strong echoes or the high 

 transmission losses produced by wakes. 



29.3.2 Effect of Velocity 



Microstructure 



A separate analysis must be carried out for the case 

 where the velocity of the water varies from place to 

 place in the medium. This is a more complicated 

 situation than the one in which the temperature 

 changes, since the fluid velocity has a direction as 

 well as a magnitude. However, it can be shown that 

 equation (5) is still applicable if the component Vx 

 of the fluid velocity in the x direction is used in place 

 of Ac. This seems a reasonable substitution, since it is 

 only the component of the fluid velocity along the 

 direction of the incident sound wave that affects the 

 propagation of this wave. 



To compute k, then, integrals of the form 



f 



sin (4:irx\)dx 



I I Vxdydz 



(12) 



must be evaluated. If the integrals over y and z are 

 computed first, it is easy to see that the entire inte- 



