94 



so that the fraction of this energy lost per cycle is 



s 



ca 



= 2 



rizT 



fp< 



(4) 



by (13) of Appendix 1. Jn the usual acoustical theory 

 the energy radiated by a simple source in one period is 



X ± 



w 



where A s AlTe., ( 4r) is the "strength" of the source. 



V'^Vmax 

 The ratio of this to (3) Is easily verified, by using (13) 



of Appendix 1, to be the same as (4). For p - 1 atmosphere, 



y= 1.4, c 3 1.5 X 10 cm/sec, (4) comes out to be ,086. 



(il) In the neighborhood of the naxinum radius let 



v;here the tir.ie t-0 is taken as the time of maximum radius. 

 The negative of the right of (1) is then asymptotically 



Cx / ^S^*)"" 



12T y^poo /a^ 

 c la 



max 



dt 



(5) 



I 





^%l£X- ^^ 



The kinetic energy is 

 2 



"-?<ll}~«^e.U(!2^/ 



and the ratio of (6) to (7) is 

 2 0^^ fB.t \ t^ 



f- 



e^^^W^ax =^-'^^^(feax) ^< 



(6) 



(7) 



(8) 



