491 



a'°;ndix 1 . 



lUTEGRATION OF EQUATION (6\ FOR ALL CHARGE WEIGHTS 



Let us suopose that we have intsgratea (6) for a soecific value of the oarameter c, 

 say c = 0.10, corresDonainj tc M = 100 lb. and for any specific value of m, say m = m, Let the 

 solution oe aenoteo Dy the suffix zsro, i.e., a , t satisfy. 



" T7T 



NOW let a = (10 c) •* a^ 



ID 

 t = (10 c)"^ t^ 



Then after suOstitution of these new variables in (i) we get 



(1) 



(il) 



l-ca-*-i2 (100 c^mlM (111) 



7^ 



It follows then that any solution a , t , of equation (6) for c = 0.10 Is transformed 

 by the relations (ii) into a solution a, t, of equation (6) with any selected value of the oarameter 

 c Out for a value of the momentum constant m = (100 c ) m. It is therefore only necessary to 

 Integrate (6) for one value of c and for a range cf values of m. 



For exaraole equation (7) for the minimum radius has oeen solved by tabulating and plotting 

 m f^r a ranue --f values cf a, and fjr c = 0.10. By relabelling the jrdlnates a,/(lO c) *'^ and 

 the abscissae m/100 c the curve becomes universally soplicable for all values if c jna hence 

 all values ..f chirge weight. Since the minimum radius Is m^re iften required in feet, a further 

 change may be made by writing <i = A /lO M where A, is the rainumum radius In feet. teking use 

 of the relation (4) between c ano M we find that 



0. 117 A, 

 (C c)-' 

 so that the orainates in Figure 1 may be also labelled with this alternative expression. 



^3 



In the same way, wfte"" calculating the vilue of the half period t, for a range of values 

 of m and fur c = 0.10 it is only necessary to relabel the ordlnates t,/(lC c/"'-' ana to relabel 

 the abscissae m/ioo c to h^ive a curve a'vi"9 'a against m for all values of the oarameter c and 

 chirge weight. The ncn-oimensi jnal half oerlod t, may be converted t; real units by using the 

 time scale factor (lO M*/j) so that the ordinates of such a curve may also be labelled 

 U.67 T,/M •', In Dlottino the curve In Figure 3 the values of t, have been multiplied by 

 IOOO/U.67 so as to remove the numerical factor and give T, In milliseconds. 



It may bo remarked that in Figures 1 and 3 real lengths and times have been divided by 

 the quantity m'". This result would have been expected if the usual system of norv-dimensional 

 units employcJ in dealing with txolosive onenomena haa been Introduced at the start instead of 

 the special non-aimensional units introcucea by Taylor. In general the non-3imensional system 

 usually emoljys: in dealing with shock-wave phenomena is inapplicable to the mctlon ana 

 behaviour of the bubble owing to the role of gravity. However, when dealing with phenomena 

 near the time of the minimum radius, when the internal gas pressure is the dominant factor the 

 approximating assumptions used above remove any explicit connection with gravity effects so 

 that this System of non-dimensional units can oe effectively employed. The effect of gravity, 

 or in other words the non-aimensional oarnmeter z,, is nevertheless present since it largely 

 determines tne parameter m. 



I 



