16 



E.;uation (6) now becomes 



/!£_ 



/(l + a)x-ax -1 



witha =P/3 P„ 



(7) 



The eraluation of Ihis integral is not possible by ordinary methods. Even graphical irethods 

 run into difficulties on account of the singularities of the integrand. Hc»ever, a sufficiently 

 accurate method is availaWe on account of the smallness of a. For the strall detonators, a has a 

 value of about 1""*, and over the early part of the region of integration it therefore plays little 

 part. Thi v-ilue of the integrand, in fact, diffi-rs from -^r^ by less th^n *i up to the point 

 X = 5. The same therefore applies to the integral, the integrand being always positive. But 

 when X varies from x = 1 to x = 5 the internal pressure of the bubble dr.ops from its initial value 

 p to % = -^ . This variation in pressure is probably larger than one is likely to be interested 

 in, the pressures at subsequent points being of little significance. Hence for practical purposes 

 the pressure variations in the early stages correspond to 



^ 2 P„ 



rr/r 



o/ P. 



-.V". 1 l^..'"' 



(8) 



or, expressing this in terms of the internal pressure p, 



I?) -■] -iVS -'] 



..; [fc 



1/4 



,5/2 



(9) 



T ime interval s between i>uLs c_s . 



Equation (?) cannot be useo to aetermine the time interval T between successive pulses. 

 To d'jtermine this it is necessary to revert to the general formula (7) and to carry out the 

 integration ov^r a compltfte cycle. In this formula the denominator of tne intcgrano 



(1 + a) X -a x'* - I 



becomes zero for two positive values of x. One of these valuss is x = 1. The other is a large 

 number, which is d'notoa by x,. The physical significance of these zeros is that they correspond 

 to the turning points of the vibrations of the bubble. When x = 1 tne bubble is in its highest 

 state of compression and when x = x^ it is in its highest state of rarefaction. Furthermore, 

 th? motion during compression corresponds completely with the motion during expansion, except for 

 sign. Henc.-;, if the integral .f egution (7) is evalusted betw.en the limits x = I and x = x^ 

 the result will be t = T/2. Hence 



T = r /^ f ' "' ^'' (10) 



'^ fo J 1 / (1 + a) X - a x" - 1 



where x, is the larger positive root of 



(l + a) X - a x" - 1 = 0. 



The approximate evaluation of (lO) is again possible by reason of the smallness of a. 

 D','tails of this -.'Valuation are given in Appendix 2. The result is 



'^A 



<3S _ ll 



(u) 



