HYDRODYNAMICAL RELATIONS 27 



amount of work on the medium. This conclusion follows immediately 

 on observing that the development of a shock front is limited ultimately 

 by dissipative processes which abstract energy from the disturbance. 

 Hence, unless the source continues to supply energy by doing work on 

 the fluid, the finite amount of energy available is reduced by irreversible 

 processes at the shock front converting it to heat, and the energy of the 

 wave must decrease. 



B. Spherical waves of finite amplitude. The equations of continuity 

 and motion for a problem involving radial motion only are 



dp . 2p dUr , dp ^ 



dt r dr dr 



dUr . dUr , dP „ 



dt dr dr 



Introducing the function a defined by Eq. (2.15) and dropping the sub- 

 script r, we obtain the Riemann form of these equations: 



(2.17) 



Eqs. (2.17) differ from Eqs. (2.16) for the plane w^ave case by the 

 presence of the spherical divergence term — 2cu/r arising from the equa- 

 tion of continuity. As a result, the propagation of the functions 

 ((7 + u) and (a — u) with velocities (c + u) and (c — u) in the plane 

 wave case does not hold true for spherical waves. It is therefore not 

 true that a — u = () after a spherical wave of compression has de- 

 veloped, even at its front. The equations can, however, be solved by 

 stepwise numerical integration from prescribed conditions at a given 

 time. If we consider a sufficiently small increment of time dt and let 

 N = {a -{■ u)/2, Q = (a - u)/2 we have from Eqs. (2.17) 



dN = — dt -\ dr 



dt dr 



= --dt-\- \dr - (c + w) dtY-^ 



J dr 



dQ =^dt+^dr 

 dt dr 



■-dt+ [dr+ {c-u)dt]^ 

 r dr 



