484 



280 



B. ARONS 



Table I. Comparison of experimental impulse values with 

 calculations based on Eq. (36). 



2„ //W* (lb. sec./in.Mb.l) 



(ft.) Experimental From Eq. (36) 



40 

 283 

 533 



1.7* 

 1.33 

 1.15 



2.0 

 1.31 

 1.12 



•III. SELECTION OF PARAMETERS 



3.1. The theory developed in the preceding 

 chapter contains three parameters related to the 

 properties of the gas in the bubble. These are the 

 adiabatic parameters 7 and ^1 and the total 

 energy e per unit mass. Although the values of 7, 

 ki, and e can be estimated roughly on the basis of 

 a priori considerations relating to the properties 

 and behavior of the detonation products, such 

 estimates do not lead directly to results of suffi- 

 cient accuracy to make the theory a means of 

 calculating reliable values of the bubble pulse 

 parameters over reasonably wide ranges of the 

 independent variables. 



In order to obtain an empirical fit accom- 

 plishing the latter purpose for T.N.T., recourse 

 must be had to the experimental results reported 

 in the preceding paper- and to radius-time curves 

 obtained from high speed motion pictures by Dr. 

 J. C. Decius and his co-workers at Woods Hole. 



3.2. In general, it is seen from Eq. (14) that 

 the non-dimensional period of oscillation depends 

 on 7 and k. Since the latter parameter is a func- 

 tion of the hydrostatic pressure, as given by 

 Eq. (10), one would expect the actual bubble 

 period {T=Ct) to deviate from the ideal inverse 

 5/6 power dependence upon the depth of deto- 

 nation. Experimental results over a wide range of 

 depth {ca. 10 to 500 ft.) indicate that no such 

 deviation is observable within the precision of 

 experimental measurements. 



Reference to Fig. 1 shows that the theory- 

 would be quite consistent with the above result if 

 the value of 7 is taken to be in the neighborhood 

 of 1.25,* since along this curve a wide variation 

 in the value of k has very little effect on the non- 

 dimensional bubble period t. This is equivalent to 



* This value is identical with ihe one obtained by Jones 

 in Britain as a result of the theoretical study of the com- 

 position of the detonation products and their equilibrium 

 during the early stages of the bubble expansion. 



making t nearly independent of depth and allow- 

 ing T to vary as Pq"^'^ as given by the scale 

 factor C. 



3t.i. Substitution of the appropriate values in 

 Eqs. (7) and (8) gives the following expressions 

 for the scale factors : 



Z.= 1.733£i(W^/2o)*, (27) 



C=0.373eHl^V2o'"). (28) 



where L is in ft., C in sec, t in cal./g, W is charge 

 weight in lb., and Zo is absolute hydrostatic depth 

 in ft. If Z is the depth of the charge below the 

 surface, Zo = Z-\-33. 



The experimental data for the first period of 

 oscillation of the T.N.T., bubble is represented 

 by 



T = 4.36Wi/Z!>'i'. (29) 



Since T=Ct, combination of Eqs. (28) and (29) 

 gives 



eH=\1.7. (30) 



It will be seen below that, for the depth range 

 which has so far been accessible, k lies between 

 0.1 and 0.3. From Fig. \, t in this range has an 

 average value of 1.483 along the 7 = 1.25 curve. 

 Putting this value of / into Eq. (30) gives 

 6 = 490 cal./g. 



3.4. There still remains the problem of select- 

 ing the third parameter, ku Fitting it to the peak 

 pressure data is the most sensitive method. 

 Expressing Pq in terms of the hydrostatic depth 

 Zo, setting 7 = 1.25, £ = 490, and combining 

 Eqs. (7), (10), and (21), 



AP,„ 



3.55(10') W' 



/J- 



R 



-(1-4^^), 



(31 



where AF,„ = excess peak pressure, lb. /in.-, 

 I^=charge weight, lb., € = bubble energy, cal./g, 

 yfe = /3Zo^-V(7-l)e\ Zo = absolute hydrostatic 

 depth, ft., i? = radial distance from bubble 

 center, ft. 



The term (l-4yfe^) in Eq. (31) represents a 

 correction factor which differs from unity by onl\ 

 three percent at depths as great as 1000 ft. It can 

 therefore be disregarded for all practical pur- 

 poses. The parameter represents a numerical 

 factor in the general expression for k, and must 

 be selected in such a way as to provide a fit to the 

 experimental peak pressure data. 



