482 



278 



A . B . A R O N S 



Following the convention adopted by Fried- 

 man,' it is convenient to transform to dimen- 

 sionless variables by using the following scale 

 factors for length and time, respectively : 



L=(3£/47rPo)', 

 C = Z,(3p/2Po)'. 

 Combining Eqs. (5), (6), (7), and (8), 



a^d' + a^ + ka~ 



' = 1, 



(7) 

 (8) 



(9) 



where a = A/L, the dot denotes derivative with 

 respect to non-dimensional time, and k is given 

 by 



k = ' 



kiPo-'- 



(7-l> 



(10) 



e is the bubble energy per unit mass of gas (or 

 original explosive charge). 



It will be noted that throughout the following 

 analysis the total energy for a cycle of the 

 oscillation will be considered constant. This is a 

 very close approximation for most of the parame- 

 ters since appreciable quantities of energy are 

 lost by the bubble only during the very short 

 intervals of time in the neighborhood of the 

 bubble minima. The treatment of the peak pres- 

 sure of the pulse, however, may be appreciably in 

 error because of this approximation. 



2.2. Since the bubble is at maximum or mini- 

 mum when d = 0, the corresponding maximum 

 and minimum bubble radii are given by the roots 

 of the equation, 



a3 + /ta-'<T-"-l=0. 



(11) 



The non-dimensional maximum radius is the 

 root near unity of the above equation, i.e., 



a.„ = [l-ifea,,r'(v-i)]i. 



(12) 



The actual maximum radius is given hy Am — ^o.^ 

 The non-dimensional minimum radius is 



^^^1/3(7- 



(13) 



2.3. Assuming the radius-time curve to be 

 symmetrical about the time of bubble maximum, 

 integration of Eq. (9) gives the non-dimensional 



period of oscillation, 



[l-a'-yfea-">-»]i_ 



-da. 



(14) 



The actual period of oscillation is given by T=Ct. 



Shiftman and Friedman have given a method 

 of obtaining this integral' with a high degree of 

 precision. Figure 1 shows a family of curves 

 giving I as a function of k for various values of y. 



2.4. Bernoulli's equation affords a relationship 

 for the excess pressure at points in fluid : 



At?/p={dWdr)-^{V^y-. 



(15) 



Here t is dimensional time and <p the velocity 

 potential which, in the case of sphericall)- sym- 

 metrical incompressive flow, is given by 



,p = A^A7R. 



(16) 



The second term in Eq. (15) is negligibly small 

 compared with the first. Neglecting this term, the 

 excess pressure is given by 



Ap = p{AWy/R = 2PoLia'd)/3R. (17) 



An expression for (a'd)- can be obtained from 

 the equations of motion ; the most convenient 

 method is the application of the Lagrange equa- 

 tion to the energy relation of Eq. (6). The result 

 is substituted into Eq. (17), yielding 



AP-- 



P(,L[a-d 



'oL\a-a (7-l)ft-| 



— a+ 



R L 3 a'-'-' J 



(18) 



2(a2d)-=ad2-3a + 3(7-l)/fea-'T+'. (19) 



When a is at a maximum or minimum, a is zero. 

 The excess pressure at bubble maximum becomes 



AP.M = 



PoLf (7-1)^1 



R l" a.u'>-' J 



(20) 



Using Eq. (13), the excess pressure at bubble 

 minimum is given by 



APn, = - 



(21) 



In Eq. (20) it will be noted that (since y is of 

 the order of 1.3, k of the order of 0.2, and a.\, 

 about 0.9) the first term predominates, the 

 second term representing a relatively small cor- 



