Appendix. 



APPROXIMATIONS IN THE PROPAGATION THEORY 

 OF KIRKWOOD AND BETHE 



An exact estimate of the errors involved in any approximate propagation theory 

 for shock waves requires either an exact theory for comparison or appropriate experi- 

 mental data. The former does not exist and the latter are treated in some detail in 

 Chapters 4 and 7. It is of interest, however, to form some theoretical estimates of 

 the nature of the errors in Kirkwood and Bethe's theory, as outlined in Chapter 2 

 and developed more fully in Chapter 4. This analysis is conveniently made in a 

 somewhat indirect manner in terms of the Riemann function Q = ^(a — u). For 

 acoustic pressures the hydrodynamical equations for the velocity potential func- 

 tion $ reduce to the wave equation 



The enthalpy and particle velocities are obtained by the relations 

 / A ON * 1 a* 



r dt 



which are also valid in the general case if <J>/r satisfies Eq. (2.23) of the text. 

 The solution of the wave equation (A.l) for an outgoing wave is evidently 



* = *0-^) 



and the first of Eqs. (A.2) may be written 



* , O 



u = ~+~ 

 r2 Co 



The kinetic enthalpy il and enthalpy co differ only by the term u'^/2 which in the 

 acoustic approximation is negligible. The Riemann a and the enthalpy oj are de- 

 fined as 



(A.3) a = 



J po P J O P 



We can therefore write, c being independent of p in the acoustic case, 



The function Q therefore becomes 



an approximation also valid for incompressive flow (c -^ <»). 



