142 THEORY OF THE SHOCK WAVE 



All of the theoretical developments are subject to some uncertainty 

 at points near the charge for want of experimental data on the prop- 

 erties of water at the extremely high pressures involved. Any results 

 for greater distances of course depend on these properties and on the 

 exact conditions in the detonation wave. A more complete and ac- 

 curate knowledge of underwater explosion pressures in general will 

 therefore depend to a large extent on measurements and calculations 

 for times during and immediately following detonation, and further 

 work along these lines is highly desirable. Work of this kind has par- 

 ticular interest in determining the energy dissipated by the shock wave, 

 as discussed in the next section. 



4.8. Dissipation and Energy of Shock Waves 



The energy in the shock wave from an underwater explosion is of 

 importance from several points of view. The energy transport at a 

 given distance from the source is a measure of the useful (or destruc- 

 tive) work which the shock wave can do. The total energy radiated 

 from the source in the initial phases of an explosion determines how 

 much of the available chemical energy remains for the later motion of 

 the gas products and surrounding products. 



Ultimately, of course, all the energy radiated in the shock wave is 

 degraded into heat by dissipative processes as the wave is propagated 

 outward. In an infinite medium, the transformation results in an in- 

 creased temperature of the water, and the least ambiguous statement 

 of the energy loss is in terms of this increased heat content. A formu- 

 lation of this kind is used to advantage in the shock wave propagation 

 theory of Kirkwood and Brinkley discussed in section 4.6. For pur- 

 poses of analysis in terms of measured quantities this approach is un- 

 desirable, and it is necessary to examine the less clear-cut relation of the 

 possible forms of energy to experimental shock wave pressures. 



A. Work and energy behind the shock front. As discussed in section 

 4.6, the energy dissipated by the shock wave at points beyond a surface 

 in the fluid defined by the position of the shock front at a given time 

 must be equal to the work done in the displacement of this surface by 

 the shock wave. Assuming spherical symmetry, the work W{R) done 

 on a surface having a radius R before arrival of the shock front at time 

 t{R) is given by 



(4.37) W{R) = 47r r 



J t.( 



r~Pudt 



t{R) 



where u is the particle velocity for the shell of radius R and iidt is there- 

 fore its displacement in time dt. This work is, however, not directly to 

 be identified with the energy radiated by the shock wave and lost from 



