approximation, in the atmosphere, for the small-scale components, according to the 

 universal equilibrium theory), <f?(k) then depends only on the vector magnitude 



e 



k = 2k sin -, (29) 



2 



and the decrement of the incident wave becomes 



/v / e 



<r(l)dl = tt 2 k 4 / cos 2 B ^ 2k sin - 

 Jo \ 2 



= 2tV / si-- ) *(s) ds. (30) 



7o V 2 K 2 / 



A similar set of results can be obtained for water. Variations of p and c will 

 arise here from variations of temperature and of salt concentration (and possibly from 

 the presence of air bubbles, in some circumstances). From data given in the textbook 

 by A. B. Wood (1932), it can be calculated that, for a fluctuation SJ in absolute 

 temperature under normal conditions, 



Sp 8T 8c 8T 



- & - 0.06 — , - « 0.80 — , (31) 

 p T c T 



and, for a fluctuation SC about the normal level of salt concentration (roughly 0.035 

 gm salt/cm 3 water) , 



8p 8C 8c 8C 



- ^ 0.025 — , -"« 0.025 — . (32) 

 P C c C 



The relative importance of all these separate influences on the scattering will probably 

 depend on the prevailing conditions, but it seems likely that temperature fluctuations in 

 the sea will often be more effective in changing p and c than fluctuations in salt con- 

 centration, and that temperature fluctuations will be relevant to sound scattering 

 primarily through their influence on c. At any rate, I shall assume so here for illustra- 

 tive purposes. 



With this assumption, we can put p 1 /p ~ and c x /c ^ 0.8 — , and the 



To 

 equation for the sound pressure becomes 



1 d'-p 7\ 



V 2 p - - — = - 1.6 — V 2 p, (33) 



Co 2 df 1 T 



which is simply the ordinary wave equation with non-uniform phase velocity. We can 

 now proceed to apply the general analysis of scattering once again. The function Q(x) 

 is here given by 



■ T 1 (x) 



Q(x) = 1.6« 2 , (34) 



T 



and the associated scattering cross-section is 



<r(l) = 1.28™ 4 i2(fc), (35) 



where Q,(k) is the spectrum of the relative fluctuations in absolute temperature 



T 1 (x)/T in the sense in which <i>(fc) and fy(k) are the spectra of Q(x) and pi(x)/p . 



418 



