14 - 2 - 



hijn pressure ana temperature. This immediately oejins to force outwards the water in contact 

 with it, a notion which is commjnicateo to a tasser txtt-nt to all parts of the surroundlnj fluid. 

 The potential energy initially possessed by the jas bubPle by virtue of its pressure is thus 

 jradually conmunic?.tco to the water in the form of kinetic enerjy. By reason of the inertii of 

 the water, this motion overshoots the point at which the pressure in the bubble is equal to the 

 external pressure of the liquid. The bubble thereafter b3Comes raretisd, and its radial motion 

 is slowed up and brought tc rest. The external pressure now compresses tne rarefied bubble. 

 Again the equilibrium configuration is overshot, and since by hypothesis there has been no loss 

 of energy to the system by radiation or dissipation, it foUcws that the bubble cones to rest at 

 the same pressure -.nd volume as at the moment of tne explosion. The physical aspects of the 

 explosion are therefore reproduced, and theoretically this process goes on repeatedly, with 

 undiminished amplitude. In practice, of course, energy is lost by acoustic radiation and by 

 dissipation, and this causes a progressive diminution of the amplitudes of the successive pulses. 



The fact that the C.R.O. records show a series of compress ional pulses with only srull 

 rarefactions is readily explained. The peak pressure of the bubble is of the order of "OOO 

 atmospheres. The maximum degree of rarefaction of the bubble "tat can occur is equal to the 

 external pr-,^ssure of the sea (say 1 atmosphere). Thus although compressions alternate with 

 rarefactions the magnitude of tha latter are Sitiall in comparison witn the forirer. 



The fact that the C.R.O. traces show tne successive compressions as isolated peaks, 

 separated by long rjgions finors tne amplitude appears to be zaro, results from the high degree of 

 asymmetry of the vibrational properties of a bubbU when the pressure variations are excessively 

 high. In such cas^s the vibrations no longer show the sinusoidal character of a bubble vibrating 

 with small amplitude. immediately after the explosion the high pressure causes a very rapid 

 expansion of the bubble. This, in conjunction witn tne smallness of the bubble, results in a 

 rapid drop in pressure which soon becomes immeasurably small In comparison with the peak value. 

 The bubble, however, moves relatively slowly when its size is greater, so that the najor part of 

 tne time interval between two successive compressions Is taken up by the bubble in moving at 

 pressures which are insignificant comparod with tne peak value. 



Math email cal dcvelopn^nt . 



The foregoing description of the expanding bubble is now fornulated mat heirat ically. 



The gaseous products of the explosion form a bubble cf radius r and pressure p , the 

 radius and pressure at any subsequent time t being r(t) and p(t), or simply r, p. The exterrel 

 pressure of the s;a is P. It is required in the first instance to determine the variation of p 

 and r with t. This is most easily obtained 'by writing down th3 energy equation for the system, 

 which is ^ot as fjllows;- 



Pptential Energ y. 



This is simply the work done by the bubble when it expanos from radius r to infinity. 



I.e. '/(r) = (p -^) «7r r^ jr (l) 



Kinetic Ene rgy. 



This Is the energy associated with the water by "eason of its radial motion, By the 

 conditions of continuity of the medium and spherical syrmetry, the radial velocity at a point 

 distant y from the c-ntre of the bubble is r dr 



y' ' dt 



The total K.E. of the medium is therefore 



i . p. 1*77 y^ ay), jr^ 2: I = 2 77pr^ r^ (2) 



