SECONDARY PRESSURE WAVES 357 



for the initial rate of expansion. In this equation, a' and t' are the 

 dimensionless variables introduced by Taylor and related to the radius 

 a and time ^ by a = La' and t = 'Vl/q t', where L = (Y/pogy'"^ and Y 

 is the total energy. Substituting, Eq. (9.4) becomes 



(9.6) P - Po = LKg/2Ty" - T a'" 



rat 



1 /a \3/5 /-4/5 



)0)i/^ \27ry r 



(1250) 



As in the acoustic approximation for compression Avaves, the pressure 

 at a given time varies inversely as the distance r in this approximation 

 and decreases slowly with time. The time dependence is the same as 

 that predicted by the complete pressure formula of the Kirkwood- 

 Bethe theory for the shock wave and initial motion of the gas sphere 

 (see section 3.8). 



It is of interest to compute the pressure obtained in this noncom- 

 pressive theory and compare it with the observed pressure-time vari- 

 ation after the initial shock front. If Eq. (9.6) is expressed in English 

 units it becomes 



/-4/5 



(9.7) (P - Po) (lb./in.2) = 1.74-10-3Li2/^^— 



r 



L and r being in feet. The scaling factor L = 0.47 F^^^ if Y is in calories 

 and it is seen that the initial pressures are independent of depth. Using 

 the example given by Taylor (107) of the pressure 14 feet from a 4.66 

 pound TNT charge gives 



(9.8) (P - Po) (lb./in.2) = 0.39^- 



4/5 



where t is time after detonation. 



A basic difficulty in comparing this result with the measured 

 pressure-time curve lies in the fact that noncompressive theory assumes 

 an instantaneous propagation of pressure to all points in the fluid. Ac- 

 tually a time of 2.8 msec, is required for the pressure wave to reach a 

 radius of 14 feet, and the noncompressive approximation is an inadequate 

 description of pressures changing significantly in this interval. A crude 

 accounting for the wave propagation is to assume that the pressure is still 

 given by Eq. (9.8) but occurs a time R/co later, where R is the distance 

 from the source. If this is done, the upper dashed curve of Fig. 9.1 is 

 obtained, measured shock wave pressures being indicated by the circled 



