SECONDARY PRESSURE WAVES 365 



where k* = 0.0607 Zo"^ for TNT, Zo being the hydrostatic depth. The 

 buoyancy of the bubble should, for consistency, be neglected also in Eq. 

 (8.25) for the translational momentum, which therefore becomes 



(9.15) fJ>(T?l 



= 



The bracketed term represents the vertical momentum acquired during 

 the expanded phase of the motion, which thus remains essentially con- 

 stant near the minimum radius, when the buoyant force of the gas is 

 small. (See section 8.5 for a more detailed discussion.) 



In order to determine the pressure, it is necessary to evaluate the 

 quantity d/dt* (a*^ da'^/dt*) , which may be written 



dt*\ dt* J dt*^ ydf'J 



It is convenient, in solving the energy equation (9.14) for da'^/dt*, to 

 eliminate the velocity dh*/dt* by introducing the constant of vertical 

 momentum s = 3^ a*^ db*/dt*. Solving Eq. (9.14) for da*/dt* then 

 gives 



(9.17) (d^y = (±)^^^.i _,*,*- 



15/4 



Substitution in Eq. (9.16) for da^'/dt'' and dV/dt'"'' (obtained by dif- 

 ferentiation) gives for the dominant term in pressure at a point r 



(9.18) 



C*2 r L2a*3 2a*s 8a*'^/\ 



2PoL* I d ( ^^ da*\ _ L*3 a* r 1 , S.s^ k* 



where L* and C* are the scaling factors for length and time defined by 

 Eq. (8.51). 



A. The peak pressure. The condition on the radius a for the peak 

 value of pressure is easily shown from this equation to be that a* have 

 its minimum value. The derivative dP/da* is given by 



3r dP 



2L*Po da"" 

 from which, using Eq. (9.17), 

 37- 



\_a*^ 2 a*6 32*a*i5/4j 



2L*P. 



dp_^ _ r/^Y . ?!! . 21 fe* "I 



