THEORY OF THE SHOCK WAVE 137 



Carrying out the indicated differentiation gives 



Pm dt Um dt R fX 



and the energy function F{R) can be written 



(4.35) F{R) = R''P„,Umi^{R)fji 



In this equation, v(R) is the normahzed integral over reduced time r 

 expressed by 



I'iR) = fjiR, r) 



, ,,p , r'^Pu , t - UR) 

 where /{R, r) = — , and r = - 



R^PmUm M 



The vakie of the integral v{R) thus normalized is an expression of the 

 shape of the shock wave, having the value one for an exponential decay 

 and two-thirds for a linear decay (sawtooth wave) . In the case of water, 

 the shock Avave is observed to be nearly exponential in form except for 

 values of R greater than several hundred charge radii. Kirkwood and 

 Brinkley therefore assign v the constant value unity for calculations of 

 underwater pressures, this being their "similarity restraint" on the 

 energy flux-time curves. A different method of applying the restraint 

 is necessary for blast waves in air; for details of this procedure the 

 original report should be consulted. 



The desired fourth partial differential equation is obtained from Eq. 

 (4.34) by use of Eq. (4.35), the complete set being: 



/^ 25) JL ^ 4- JL ^^ 4_ ?!f!!^ = _ R'^R^'^rn 



Um dt Pm dt R F{R) 



dUm , ]:^Pni ^ Q 



dt po OR 

 PodR pc" dt R " 



dUm , jjdjLrn _ Q dPm Q dPm ^ ^ 



dt dR Po dR poU dt 



With this set of equations, it is possible to solve for the derivatives 

 dPm/dR, dPm/dt in terms of coefficients depending only the shock front 

 pressure Pm and radius R and hence obtain an ordinary differential 

 equation for Pm in terms of R : 



