23 



DC *ritt'.'n 



r . ..1,2 





Inserting actual values it is tounJ that, after 2 fi seconds 

 — = 1.018. 



Tn^re is thus very little expansion in this time interval, so that it ,7.akes little difference whether 

 the explosion jccurs at the Deginninj cf this interval or is distriouted over it. 



(c) Th e radiation ^ f acustic etierjy . 



It was earlier assuined that there was no radiation of sonic energy. Taninj the solution 

 obtained on this assumption as a first approximation, it is now possiDlo to calculate what energy 

 is actually rddiatefl wnen account is taken of the finite speeo of sound, c. 



Tne instantaneous flux of sonic energy per unit area per unit time at a ran^e a is 



SuDstitutina f or n from equation {2K) this becomes -^ — T~K • Integrating over the surface of a 



pc a r 

 sphere of radius a, the rate of tmission of energy from the bubble becomes 



/Oc r pc X 



The energy emitted in the cours^i of the first pulse is therefore 



= 1 



(28) 



where x, h^s the same significance as in equation (lO). 



Puttinj — = X in equation (8) ind di f ferent iat inj, jivis 

 



2t = r /^m ^iji 

 <= / 2 p^ /7- 1 



and substituting this in (28) jives 



r 3 r, 3/2 



In equation (29) the major part of the intejral arises from values of x near to unity. This fact 

 justifies the use of equation (8) over the range of integration x = 1 tj x = x, for equation (8) 

 is valid only over the initial part of this interval (x = 1 to x = 5). The srror introduced on 

 this account is °xtremely small, and for the same reason it is juitc legitimate, and more convenient 

 to write the 1 imits of integration in (29) as x = 1 to x = co. Then 



r i ^ ill 

 w = 27t/2 



cp"' \ xVx- 



r? n3/2 



(30) 



This is now to be compared with the total energy w available at the moment of the explosion. 

 Takingly = «/3, this latter by virtue of equation (l«) Is *77 r ' p so that the fraction of the 



