HYDRODYNAMICAL RELATIONS 31 



This has the solution $ = $(/) by integration, applying the condition 

 that G — ^ as r — > "^ . In this case, therefore, the function rQ, = d^/dt 

 is propagated outward with infinite velocity. If the disturbance is 

 sufficiently weak, terms involving u^ are negligible and the wave equa- 

 tion 







has a solution of the form ^ = ^{t — r/co) for an outgoing wave, where 

 Co is evaluated for the undisturbed medium. In this case then, G = rQ, 

 is propagated with a velocity Co. 



The limiting ways in which G is propagated make natural the as- 

 sumption that, in the case of finite amplitude, G is propagated with a 

 variable velocity c, as expressed by the relation 



(2.24) 



= (-) 



The function r{G, t) may be thought of as a series of curves in the 

 r, t plane for various assigned values of G. It is reasonable, from the 

 limiting cases considered, to assume that the slope c of these curves for 

 an outgoing wave is finite and positive, and that these curves for the 

 region behind the shock front do not intersect. With these assump- 

 tions, the function G(r, t) may be expressed as a function of a single 

 variable rir, t) 



(2.25) G(r, t) = Ga{r) 



the functional form of Gair) being unrestricted except for the require- 

 ment that T(r, t) be a single valued function of G(r, t). With this con- 

 dition, Eq. (2.24) for the propagation of G may be written 



\dt)r 



c{r, r) 



If the function G is to be a useful one, we must be able to relate its 

 value at any point to conditions on the boundary of the gas sphere. By 

 our assumptions, the function G(r, t) has, at some earlier time t\ the 

 value G[a{t'), t'] at the gas sphere boundary, for which r = a{t'). The 

 time required for G to be propagated to the point (r, t) is then 



