488 -2- 



The reason fur this assunijtion is that near the time of the minimdm raoius the integrand is v^ry 

 small comcareo to its value Jurioy ths lirge jart of the oscillation when the ouDble is large., 

 Accordinjly equation (2) is reolacea near the tin;e of th^ minimum radius by 



Methods of comojtlnj m in the oressnCG of various surfaces or in oosn wster are oescriDed in 

 referendes-2, 3, and 4, It should Ds realised that in what follows it does not matter whether 

 t-his momentufTi arisos solely as the result of the Duoysncy fcrces acting en the BubDl e :r frcm 

 the attraction :r reoulslcn exerted by rigid cr free surfaces. 



The secona assumotion made is that the term containing z in equation (l) is negligiDlc- 

 comcared to the otner terms during the short ocrioO near the minimum radius when the ciressure 

 culse is mainly croQuc^a. The r^asonaDleness of Doth these ^ssumotions Is Oest demonstrated 

 Dy comcarison :f tho results arising from this 23:roximate treatment with the results calculated 

 by numeric-*! integration of the full di ffcrent i;il equations. This comoaris:.n will De made below. 



with these two assumotlons equation (l) reduces to 



The nlnimum radius a, follows at once from (6) by setting the quantity under the square root 

 equal to zero, giving 



^- .{ (1 - c. "-) = .^ (7) 



Equation (7) is not easily 5clvea by successive aacrcxiovit ions. Moreover, since it contains the 

 oarameter c *hich OeosnOs 3n cnargs weight, ana the momenturr; c-nstant m it is not easy to Dortray 

 it grashically. However it is sh:wn in Jocendix i that by 2 suitaole change of variables a 

 universal minimum raaius curve may de clotted, 5uch a curvt has been olottea in Figure 1, 



The p res su fe pul sc 



In tho B'.rnoulli equation for the excess oressure p in the water the term falling off 



as the inverse first aow.T of tne oistance will oreflominate over other terms with higher inverse 



cowers excect quite closs' to the Duobl 5, At 3 distance r from the centre of the bubble this 

 term is 



- i ^ {.' ; ) 



r at 

 Usinj (6) in this oxoression, and after som? reduction, we get 



ro = ~S [1 - , ca-* . iiL^f ] (a) 



Thus c is given by equation (8) in terms of the radius a with c and m as parameters. 



As it stands (8) is cf little use since we nave no analytic exoression for a in terms of the 



time t. However, it nay be Shown that the maximum oressure arises at the instant of the 



minimum radijs and ty substituting from (?) into (b) we get the following exoression for the 

 maximum pressure d . 



'■^n, = —^ (1 - i c.,-') (9) 



(it is conv.'nivnt to have the :ris3ure exoresseo in lb,Cin.^ »na the distance in feet and this 

 will result if tne quantity on the right hand side cf (3) is multiplied by U3,4 M^ ) . 



Thf above formulnO w^r; derived in reference 2 and ire reoeatec here f;r convenience. 



In order tc fAke use cf (8) to obtain the shaoe of ths oressure oulse as well as its 

 oeak value it is ncctssary to obtain the radius a as a function of time t, i.e., it is-necessary 

 to integrate (6) In the neighbourhood of the minimum radius. There seems no simple way, either 

 of integrating (6) by exoansion in series, or cf clotting or tabulating the resulting solutions 



