156 



Hence, i' when s is odd, we write 



^n.s = ^s9j^'^«U2.s^'^n-''' J„(z). ^ ' a - a^ (59) 



where g (2) and j (z) correspond to the two fundamental solutions in (50), the continuity of /3 

 at a = a requ i res 



B = B .,, {») 



o,s o,s+l 



Also 



'^" ' Vs ^'n,'^) ^ Vz.s^^'^"^' JV^'- 2/(i,-a) Jj^) > 



n.s 

 d a 



and therefore 



Similarly 



so that the continuity of ^ at a = a^ requires 



^iCZ.s ' ■ ^/2,s*l (61) 



In the same way the continuity cf /S and p at a » 1 determines the relations between 

 P and /S 5* , *''en s is even. Writing 



^n,sM= \,s*l ^1'') *''i/2,sm'^*'^-'' Jl'^' 



o,s 



(42) 



(63) 

 («*) 



The solution when PjQ » jo'^, n = s 



The case when P/Q is large Is of most interest in connection with underwater explosions, and 

 the above result has thsrcfore been evaluated for the sec,nd harmonic term, with P/O = 10 . The 

 second harmonic S, includes (among others) the deviation which deforms a sphere Into a prolate or 

 oolate spheroid. It will be assumed that this deformation exists at the Instant of the explosion, 

 and that its velocity Is Initially zero. 



When P/O = 10 , the value of the maximum radius is found to be 



a^ = 30,73, (65) 



and the period of pulsation of the bubble is 



T = 0,056 a (T In spc. and a in cm.) (66) 



These values ar? of the fame order as those observed for T.N.T, and similar explosives. 



When 



