485 



THEORY OF GAS GLOBE OSCILLATION IN UNDERWATER EXPLOSIONS 281 



Measurements of AP^ for the first bubble pulse 

 in deep water under conditions of very small 

 bubble migration give values* of about 1200 

 Ib./in.- at 11-''/^? = 0.352. Using this result in 

 Eq. (31) gives ^ = 31.8, and the general expression 

 for the peak pressure becomes 



AP„ = 3450 — {\-ik'), 

 R 



and k is given by 



yfe = 0.0552Zn>- 



(32) 



(33) 



IV. APPLICATION TO OTHER MEASUREMENTS 



4.1. In the preceding section, experimental 

 period and peak pressure results were used to 

 determine the values of the three arbitrary 

 parameters appearing in the theory. It is now 

 necessary to ascertain whether the same parame- 

 ters will also fit the rest of the available experi- 

 mental data, consisting of (a) the maximum 

 bubble radius, (b) the time and corresponding 

 bubble radius at which Ap = 0, and (c) the posi- 

 tive impulse in the first bubble pulse. 



4.2. High speed motion picture measurements 

 of T.N.T. bubbles give the maximum bubble 

 radius as 



^.„ = 12.6(lF/Zo)*, (34) 



where Am is expressed in ft. This represents the 

 average of a large number of measurements over 

 a wide range of depths (ca. 100 to 600 ft.). The 

 accuracy of the measurement is believed to be 

 about ±2 percent, and within this scatter there 

 seems to be no systematic variation with depth of 

 the numerical constant 12.6 in Eq. (34). The 

 experimental non-dimensional maximum bubble 

 radius, a^, is therefore constant and equal to 0.92 

 if « is taken as 490 cal./g in calculating the scale 

 factor. 



The theoretical values of a^ obtained from 

 Eq. (12) vary from 0.94 to 0.90 for the range of 

 depths cited above. This predicted variation is 

 not confirmed experimentally, but since it is 

 small and of the order of the magnitude of the 

 experimental error, the agreement between ex- 

 periment and the theoretical fit can be considered 

 quite satisfactory. 



4.3. The pressure-time curves- which yielded 

 the value of AP„ used in the preceding section, 



also provided measurements of the time (meas- 

 ured from the shock front) at which the excess 

 pressure dropped to zero during the initial bubble 

 expansion and the time at which it returned to 

 zero from the negative phase as the bubble 

 proceeded to collapse. When compared with 

 radius-time curves obtained from high speed 

 motion pictures, these data show the non-di- 

 mensional bubble radius ao to be 0.62 when 

 Ap = 0. Within the precision of measurement, 

 this value appears to be independent of charge 

 size and depth of detonation. 



The corresponding theoretical value of ao is 

 obtained by solving Eq. (22) using 7 = 1.25 and k 

 as given by Eq. (33). It is found that ao is very 

 insensitive to variation in the depth and has 

 values of 0.62 to 0.61 over the range Zo = 83 to 

 533 ft. This is in excellent agreement with the 

 experimental observation of 0.62. 



4.4. The positive impulse delivered in the first 

 bubble pulse can be computed from Eqs. (26) and 

 (26a), providing an adequate assumption can be 

 made concerning the magnitude of the scale 

 factors L2 and C2. Both of these factors depend 

 upon 62*, and £2 is known to be less than ei, owing 

 to the radiation of acoustic energy associated 

 with the emission of the bubble pulse. If period 

 measurements are adopted as the criterion, one 

 would expect as a first approximation that 



(«2Ai)* = r2/ri. 



(35) 



Experimental results give T2/Ti = 0.72 as an 

 average. 



Thus Eq. (26) can be rewritten 



//W^»=11.9Zo-"«(FFV^)(l-l-59/t)i, (36) 



y having been set equal to 1.25 and ao to its 

 average value of 0.615 as determined above; the 

 assumption is made that to a first approximation 

 (a^d) 01 is equal to (a-d) 02. 



In Eq. (36), //IF' = reduced positive impulse 

 (lb. sec. /in.- lb.*), Zo = absolute hydrostatic depth 

 (ft.), W^= charge weight (lb.), i? = radial distance 

 from bubble center (ft.), and /fe = 0.0552Zo^-'. 



A comparison of experimental values^ of I/W^ 

 with those calculated from Eq. (36) is given in 

 Table I for various depths Zq. It will be noted 

 that the agreement is quite satisfactory. 



4.5. From Eq. (13) it is also possible to make a 

 theoretical prediction concerning the minimum 



