lU THEORY OF THE SHOCK WAVE 



There are some experimental departures from similarity which may be 

 the result of delayed reaction or "afterburning," but for the most part 

 this effect is probably insignificant. 



The most important phenomenon in which the principle of similarity 

 as here stated is not valid is in the later history of the explosion after 

 emission of the shock wave. The laws of motion of the water and the 

 gas sphere, as developed for calculation of the shock wave, neglect any 

 external forces or boundaries. This is a perfectly legitimate procedure 

 in view of the large internal forces acting in the times of significance, 

 but is no longer so when the internal forces and accelerations have be- 

 come weakened and mass flow of the water has established itself over 

 large regions. The more important nonscaling factor under these con- 

 ditions is gravity, because it is always present, but under many condi- 

 tions boundaries, such as the surface and bottom, have an important 

 effect on the mass flow and motion of the gas sphere. 



The sphere of gaseous products expands very rapidly after its for- 

 mation and soon occupies a volume much greater than that of an equal 

 mass of water. As a result of this hydrostatic buoyancy the bubble can 

 be thought of acquiring vertical upward momentum in addition to 

 momentum associated with radial motion of the water. The energy 

 associated with the upward rise must come at the expense of other forms 

 of kinetic and potential energy and is more important for large charges 

 in which the time scale of the motion is longer. If the vertical displace- 

 ment of the bubble could be neglected and noncompressive flow as- 

 sumed, the motion of the gas sphere would scale according to the prin- 

 ciple of similarity, as shown in section 8.2. These assumptions are, 

 however, not legitimate except as a crude first approximation, and in- 

 clusion of gravity effects makes the principle of similarity, as here 

 stated, inapplicable. Other types of scaling law which permit a non- 

 dimensional approximation to the laws of motion for the gas sphere 

 have been worked out by G. I. Taylor and will be described in section 

 8.5 (see also section 8.9). 



4.2. The Detailed Evaluation of Shock Wave Propagation 

 (Kirkwood-Bethe Theory) 



In Chapter 2 it was shown that the propagation of a spherical shock 

 wave could be descril^ed in terms of the kinetic enthalpy 12(r, t) defined 



by 



(4.2) nir, /)=:^ + ;i = co + ^ 



the contributions to 12 from entropy changes behind the shock front 

 being neglected. The utility of this approach lies in the fact that 12 can 



