III-36 



Hence, if we substitute (III-32) in (III-29a ) and ignore terms of order (ka) , 

 we find that the internal pressure Ap' is proportional to Ap: 



.p..!lE.ii!L.p=(^)= £^.p „„_33, 



U)° p a" " ' ^ 



Since we know the variation in internal pressure, we can calculate the variation in 



4 3 



volume of each bubble . The undeformed bubble has a volume Vj^ = — tt a ; when 



its internal pressure is changed adiabatically from p to p + Ap' , its volume 

 must change so as to keep pV^ constant, so that: 



b Y P 



The total change in the volume of air A V is therefore (using Equation III-33) 



NV Ap' 3g(a)NV 4TTnVQ g(a) 



AV - NAV = = Ap (III-34) 



a b p Y uu^p a^' U)% 



We can now evaluate the complex velocity of propagation c according to (III- 30). 

 It is a little easier to compute its inverse: 



P AV P ^V„, P ^V„ 



- 3 _ '^o tA V _ o_ W _ o_ f_ 



^ " " V Ap ' V Ap " V„ Ap 



^ O "^ ^ 



The first of the two terms on the right is seen to be the inverse square of the 

 sound velocity in pure water, c^^ ; the second term may be simplified by using 

 (III- 34). Therefore, we find 



c-^ = c-^ + iHH-lii) (III-35) 



S-7001-0307 



