- ^ - 509 



Di scussion - Compari son unth Theory . 



(a) Radius - Time Curves :- In Figures 1 ana 2 the mean radius of the bubble - defined as 

 the radius of a sphere having the same volume as the bubble - is plotted against time. For comparison 

 with experimental data, values of the radius as calculated by the Nautical Almanac office have been 

 plotted in Figures i and 2. These calculations refer to a non-dimensional depth z = 7.2, the bubble 

 being assumed to remain spherical, and account has Been taken of the presence of a free surface 0.6 

 units above the charge. These correspond to a l-oz. charge with a bubble energy of kiio calories/gm. 

 at a depth of ) feet. " The standard length for this case is 5 feet and the standard time 0.391 seconds 

 yielding a value of 77 milliseconds for tne period. This period is so close to the observed average 

 period that the theoretical results have Been plotted without any adjustments of scale. 



It appears fron Figure 1 that there is a discrepancy somewhere in th.Tt the periods agree while 

 the observed maximum radius exceeds the theoretical value by about e%. If the energy of the motion be 

 calculated from the observed maximum radius, (assuming that the fraction of th" energy left in the gas, 

 amounting to about 15$, is the same as in the theoretical case) a figure of 550 calories/gm. is obtained. 

 The energy calculated from the period assuming the correctness of the theory, is only uuo calories/gm. 



In assessing tne cause of this discrepancy it does not seem likely that the maximum radius 

 measurement is more than z% in error, while the measurement of the average period might be as much 

 as 3* in error. These two might conceivably contribute an error of 15* in the energy, in the worst 

 case where the errors act in the same direction. 



It is possible that the theory of the effect of the free surface is somewhat in error for a bubble 

 as close to the surface as this, since the theory is only an approximation which neglects cubes and 

 higher powers of the reciprocal of tne distance of the free surface. According to the approximate 

 theory used the presence of the free surface reduces the period from 86 tc 77 milliseconds in this case; 

 a large correction which if itself in error might account fcr some of the discrepancy. It has also to ' 

 Be remembered that the effect of the rigid b:ttom tt a distance jf 6 feet 6 inches from the bubble has' 

 been neglected in the theoretical calculations. This, however, wculd raise the theoretical period to 

 about 81 milliseconds and increase the size cf the discrepancy. 



It is further just possible that there is a considerable energy loss during the contraction 

 stage, though most of this would have to occur while the bubble is large in order to reduce the period 

 to such an extent. it is estimated from the ratio of the first to the second period that about 60t( 

 of the bubble energy is lost between the first and second oscillation, but it seems.more likely that 

 most of this loss occurs near tne minimum when the bubble is moving most rapidly in a vertical direction. 

 The data plotted in Figure 2 bring out the quite sharp energy loss as indicated by the much slower 

 expansion after the minimum, than contraction before the minimum. 



(b) The Minimum-Radius-. - Energy Considerations:- According to the theory of a spherical 

 bubble the minimum radius should be 2.01 inches when the energy in the gas is 81* of the total, the 

 redlining i9J being kinetic energy in the vertical motion of the water. The observed mean radius at 

 the minimum, based on the volume, is 2.7 inches. This puts the energy in the gas at 6Ul, assuming 

 that the gas adiabatic is substantially that of the theory. If one assumes that the effective volume 

 of water moving with the bubble is half the volume displaced (as is true for a spherical bubble and 

 streamline motion) the kinetic energy of vertical motion w:rks out at 1.9? of the total - taking the 

 observed maximum velocity as about is5 ft. /sec. 



However, the shape of the bubble at the minimum is more nearly hemispherical with a radius of 

 3.9 inches (see Figure 8d). |f one thus assumes that the effective volume of the water moving vertically 

 is one half that displaced by a sphere of radius 3.9 inches the kinetic energy at the minimum works out 

 at 5,6J of the total. (In fact the effective volume of water moving may be even greater since a 

 hemisphere moving rapdily through water is likely to drag with it a volume of dead water lying 

 Immediately behind its flat surface in addition to the water circulating round it). This brings the 

 energy in these two phases to about TOSS of the total. It is not unreasonable to suppose, therefore, 

 that of the observed loss of energy between the two cycles of aBout 601 (based on the ratio of the 

 first and second periods) half is lost just before the minimum, the other half just after the minimum 

 when the vertical velocity is high. 



