- 15 - 



159 



The graph of B as a function of a during the first two pulsations is easily constructed from 

 Taoles I and II. This is shown in Figure u. It will De seen that the variation of /3 during the 

 second pulsation is gener.illy of a similar nature to that during the first pulsation, but the sign 

 Is changed and the amplitude is increased. 



Hhen /3 is regarded as a function of t. It should De expressible in the form (l5). It is 

 easily verified that for any function of this form 



cosh \ T = \[^ [\* ^^) * fi (t) ] //3 (t + T), 



so that \ is determined by any 3 values of /fl separated by intervals of T. Taking the values of 0, 

 T and ZT as given by A in Table II, we find 



cosh \ T = - 2.06, 



X.T 



-3.95, or \ = (1.38 t i ttI/T. (72) 



This shows that eventually (if the pulsations could continue) the perturbation /5 increases 

 four-fold in absolute magnitude and changes sign during each complete pulsation. In this sense, 

 therefore, the perturbation is unstable. 



At any value of a the departure of the bubble from sphericity is measured, not by /?, but 

 by /3/a. This implies that the non-sphericity greatly decreases during the expansion. At the end 

 of the first expansion we have ,fl/a = - .443/30.7 = - .015, or about 1.5S of its value initially, and 

 of opposite sign. 



When !^/a is regarded as a function of t, it decreases very rapidly at first, due to the rapid 

 initial Increase of a. It remains small and changes very slowly during the relatively long period 

 that the bubble is large, but it Increases very rapidly (In absolute value) when the bubble approaches 

 its minimum again. These changes are illustrated In Figure 5, which shows /3Ai as a function of t 

 during the first two pulsations. It is evident from this figure that any small disturbance of the 

 bubble when it Is larje (e.g. due to gravity) would lead to very great changes in Its shape when it 

 contracts lijain. 



References . 



(1) Lamb, Phil. Mag., 45, 257, 1923, 



(2) autterworth, "Report on the thecretical shape of the pressure tim!.' curve and on the 

 growth of the gas-Oubble", 1923. 



(3) Willis, "Underwater explosions. Time interval between successive explosions", 1941, 



(4) Herring, C4 - Sr20 - 010, N.D.R.C., 1941. 



