29 217 



Because the energy depends on the square of the quantity p - p^^, 

 radiation will occur almost exclusively near the peak of the wave, provided 

 the amplitude of oscillation is large. At large distances, furthermore, the 

 term pv'^/Z may be dropped. Hence, for y = U/3, Equation [42] may be employed 

 for p, or, in terms of i = R/R^, x, = R^/Rg, 



From Equation [35] 



Ro 1 

 P - Po = Pn — 



l/ Sp „ x^dx 



dt = y -A- R 



2P» '"" (Cx - X* - 3)2 

 Hence, approximately, for y = 4/3, 



^ ^ ^1 -^ x2(Cx - x^ -3)7 



in which C is given by. Equation [36]. 



If the amplitude is large, x* may be dropped and C may be replaced 

 by 3/xi, as in obtaining Equation [37]- In integrating up to a large x, fur- 

 thermore, the limit can be replaced by oo without much error, because of the 

 rapid decrease in the integrand. Hence, if y = 4/3 and the amplitude of os- 

 cillation is large, the energy emitted during a compression and subsequent 

 re-expansion has the approximate value 



«, = ^ (^ P.* 4 2/ V^— r 



C P X,f / x2(l - X,)2 



The Integral equals rr/(2xj^'M. Hence 



^,= -^-P-^ Po2 i?,f(^) [47] 



or, if we also eliminate p by means of Equation [34a], 



For comparison, the total energy of oscillation E is equal to the 

 kinetic energy in the water at the Instant at which the radius R = i?,,, since, 

 If the water were suddenly arrested at this instant, the sphere would remain 

 in equilibrium. As R decreases to its minimum value, R^, this energy, to- 

 gether with the work done by hydrostatic pressure as the radius decreases, 

 becomes expended in work done in compressing the gas. Hence, by Equation [27] 



