THEORY OF THE SHOCK WAVE 135 



The fourth relation necessary to obtain the desired ordinary dif- 

 ferential equations is determined by Kirkwood and Brinkley from the 

 observed energy flux-time curve of the shock wave at a suitable distance 

 from the source. The reason for choosing this property of the pressure 

 wave lies in the fact that the work done by the shock wave after it passes 

 a point R is ultimately dissipated as heat increasing the internal energy. 

 This nonacoustic decay of a spherical shock wave and the loss of avail- 

 able mechanical energy are thus closely related. 



From energy considerations the energy-time integral F{R) at a dis- 

 tance R from the source can be expressed in terms of the peak pressure- 

 distance curve integrated over distances greater than R, the relation 

 being 



(4.29) F{R) = C r^Pudt = f^ro'poh[P{ro)]dro 



J t(R) J R 



The first integral is an expression of the work-energy principle, P being 

 the pressure in excess of hydrostatic pressure and udi the displacement 

 of a fluid element in time dt, and the integration is performed at con- 

 stant R. In the second integral h{P) is the increment in enthalpy, de- 

 fined in section 2.4, for a fluid element through which the shock wave 

 has passed. 



The result Eq. (4.29) is obtained by considering the work Wo done 

 by the source of initial radius a^. The passage of a shock front through 

 a fluid element leaves it in a state of higher entropy and internal energy, 

 from w^hich it returns adiabatically to hydrostatic pressure Po. The 

 total work done may be resolved into the sum of two terms : the increased 

 internal energy of the fluid at pressure Po within a sphere of radius R, 

 and the work done on this spherical surface. Thus we have 



(4.30) TF« = 47r f Poro^E[Pm{ro)^ dn + iir f" r'~'(P + PJ udt 



J ao J t{R) 



where E[P„i(ro)] is the increase in internal energy per unit mass of 

 fluid and the time integral is carried out for the volume element initially 

 at R. The term involving Po in the time integral gives the product of 

 Po and the volume displacement of the fluid element initially at R. 

 This displacement is evidently the sum of the outward volume displace- 

 ment AVg of the inner boundary of the fluid (i.e., the expansion of the 

 gas sphere) and the displacement of the volume of fluid initially between 

 the shells of radii ao and R to shells of radii a', R' . It can be written 



47r f* PoT'^udt = Po^Vg + 47rP.rf rhlr - f To'drA 

 J t(R) iJ a' J ao A 



