HYDRODYNAMICAL RELATIONS 



45 



sures with numerical values from the equation of state and Rankine- 

 Hugoniot conditions in Table 2.3. It is seen that the approximate equa- 

 tions give reasonably good agreement with the more accurate calcula- 

 tions, and therefore form a good basis for development of propagation 

 theory. The velocity c + a which Kirkwood and Bethe use as their 

 fundamental propagation velocity behind the shock front is seen always 

 to exceed the shock front velocity U. This is what the argument of sec- 

 tion 2.3 led us to expect, and makes plausible the result that the out- 

 ward motion of a shock wave involves a progressive overtaking of the 

 front by the disturbance behind the front and dissipation at the front 

 which limits its rate of advance.^ 



2.7. The Thickness of Shock Fronts 



The Rankine-Hugoniot equations are universally employed to 

 represent the conditions at a shock front. Since their derivation de- 



Table 2.3. Comparison of approximate expressions for shock front parameters with 

 numerically computed values. 



pends on the assumption that a shock front can be regarded as a dis- 

 continuity, it is important to examine whether a theoretical estimate 

 of the shock front thickness is consistent with this assumption. A 

 number of writers have made such estimates assuming that the mech- 

 anism of dissipation in the shock front could be represented by the 

 macroscopic shear viscosity of the fluid. The common result of all 

 such calculations is that they predict thicknesses of the order 10""^ to 

 10~^ cm., which are of course completely negligible. It is evidently 

 meaningless to ascribe any physical significance to lengths of molecular 

 magnitude obtained by macroscopic concepts beyond the fact that they 

 are so exceedingly small. The failure of experimental measurements 

 to yield more than an upper limit set by the limitations of the measuring 

 equipment constitutes additional adequate evidence that the physical 

 thickness of the shock front is not important in the analysis of shock 

 wave propagation. 



Of the various theoretical discussions which have been given, those 

 of Rayleigh (90) and of Taylor and MacColl (30) should be mentioned. 



^ A more general discussion of shock wave velocities for an arbitrary equation of 

 state has been given by Bethe (8) , who shows that very generally Co < U < c. 



