376 SECONDARY PRESSURE WAVES 



We have, however, that 



E(a) = 



47r Pa' 



3 7-: 



and 



E{a) 





since ^(a) = Y, there being no kinetic energy when a = a. Sub- 

 stituting in the right side of Eq. (9.26) gives 



AF = Energy radiated 



Co \2Tr po/ a J L \«/ J\«/ 



The Hmits on the integral should strictly be from the first maximum 

 radius am to the next maximum to get the total energy radiated. In the 

 noncompressive theory a is symmetrical about the minimum radius a 

 and we can therefore take tmce the integral from am to a. The inte- 

 grand is large only near a = a and the limit am can be replaced by 

 a = CO without appreciable error. Letting a/d = x then gives 



Co " V^irpoJ 



/: 



[1 - X-2(7-l)]l/23-l/2-37f/:r 



which can be evaluated in terms of gamma functions to give the frac- 

 tional energy loss 



(9.27) 



AY 

 Y 



2V6 





p_iii 



Capo 



1/2 



where the relation Y = (47r/3)P^aV7 ~ 1 has been used. 



This expression should also represent the energy flux through any 

 sphere drawn around the bubble, and Herring's result, Eq. (9.27), has 

 independently been derived by Willis (121) in this way. Willis cal- 

 culates the energy flux using the acoustic expression relating excess pres- 

 sure and particle velocity u = (P — Po)/poCo, which gives for the energy 

 flux across a sphere of radius r 



/ 



AF = iiriMP - Po)dr 



^5 C (P - p^ydi 



PoCo J 



Willis' approximation in evaluating the integral is that the pressure P 

 ''6 given by the noncompressive tlieory; substitution of P — Po = (po/r) 



isg 



