188 



EXPLOSIONS AS SOURCES OF SOUND 



the solution obtained for one quantity of explosive 

 will be valid for any other quantity if the scales of 

 rj, and t are changed in proportion to ro. The full curve 

 of Figure 7, taken from a report issued by the David 

 Taylor Model Basin ,^2 shows the form of the radius- 

 time curve obtained from integration of equation 

 (28), using a function G(ri) comparable with that 

 which would obtain for a charge of high explosive; 

 while the dotted curve is the one which would result 

 from equation (28) assuming the same maximum 

 value of r-j but setting G = 0. Several scales are given 

 appropriate to several sizes of charge. The variation 

 of Ti with t near the minimum of the contraction is 

 too rapid to show on the scale of the figure. It would 

 not be worth while to show this portion in greater 

 detail, however, since, as will be explained presently, 

 the motion of the bubble in this stage is strongly in- 

 fluenced by gravity and other asymmetrical factors, 

 and also, although less strongly, by the finite com- 

 pressibility of the water. These effects prevent the 

 present simple theory from being even approximately 

 correct near the minimum. When r^ is greater than 

 three or four times the minimum radius shown in 

 Figure 7, Gir^) becomes small and the motion is 

 practically the same as it would be if there were no 

 gas in the bubble at all, as can be seen from the 

 agreement of the dotted curve with the full one. For 

 this portion of the curve a change in paj is equivalent 

 to a change in W combined with a change in time 

 scale; thus, over most of the period of the oscillation 

 Figure 7 applies not only to charges of different sizes, 

 but to charges at different depths, provided suitable 

 time and radius scales are used; these scales can be 

 deduced from equation (29) below. 



The period of the motion, in the approximation 

 neglecting gas pressure, is easily deduced from equa- 

 tion (28) and turns out to be 



U = 1.135p=p^"TF= = l.S2^p'p^'n max (29) 



where , _ / 3TF V 



'b max \ . / 



\47rpa,/ 



and is the radius of the bubble at its maximum size. 

 This expression, in spite of its neglect of gas pressure 

 and the other effects to be discussed later, has been 

 found to agree with measured values of the period to 

 within a few per cent, provided the explosion takes 

 place in open water well away from bounding sur- 

 faces, which exert a perturbing effect discussed later. 

 With the same proviso, the variation of the period 

 with depth and size of charge agrees with the ex- 

 ponents in equation (29) to within the accuracy of 



measurement. The measured values of the bubble 

 period are found to be reproducible from shot to shot 

 to within a per cent or less.^''^^ 



The simple theory just outlined ignores the com- 

 pressibility of the water and any influences which 

 may make the motion asymmetrical. The compressi- 

 bility of the water is important near the minimum 

 of the contraction, when the pressure in the bubble 

 is very high. During this stage of the motion the 

 compression of the water near the bubble initiates 

 a pressure wave which travels outward as an acoustic 

 pulse. This is known as a "secondary pulse" or 

 "bubble pulse," to distinguish it from the original 

 shock wave. The acoustic energy carried away by 

 this secondary pulse is usually small but may, under 

 exceptionally symmetrical conditions, be apprecia- 

 ble compared with the total energy W of the oscilla- 

 tion. Loss of energy through this effect and through 

 turbulence has the consequence that the next oscilla- 

 tion, although conforming generally to the theory of 

 the preceding paragraph, is of lower ampHtude than 

 the first one, since it has an energy Wi which is 

 smaller than W. Energy is radiated in a similar man- 

 ner at each succeeding contraction. 



Several causes may act to prevent the motion from 

 having true spherical symmetry. Most important of 

 these, because it is always present, is gravity. The 

 bubble, being buoyant, tends to rise; the rate of rise 

 is limited by the inertia of the water around it. The 

 rise is usually rather slight during the first expansion 

 of the bubble, but as the bubble contracts again the 

 rise is enormously accelerated and may result in a 

 large portion of the total energy W being retained as 

 kinetic energy in the water at the time the radius of 

 the bubble is a minimum; since this energy is not 

 available to compress the gas, the minimum radius 

 of the bubble will not be as small as it would be if 

 gravity were absent, and the secondary pressure 

 pulse will be correspondingly weaker. Another effect 

 of the rapid rise is to produce turbulence in the con- 

 tracted stages; this turbulence dissipates energy and 

 is probably the most important factor in the damping 

 out of the oscillations. 



This rapid rise in the contracted stages has been 

 the object of many theoretical studies. ^-•^^~" The 

 explanation of the phenomenon rests on the fact that 

 a spherical cavity moving through a fluid possesses 

 an "effective inertia" equal to half the mass of the 

 water it displaces. The buoyancy of the gas bubble 

 causes it to acquire an ever-increasing amount of 

 vertical momentum, and to conserve this momentum 



