HYDRODYNAMICAL RELATIONS 23 



Ur = dr/dt = {a'^/r^)ua. Thus, in the Umit of small pressure differences, 

 the afterflow velocity is simply the velocity of noncompressive flow 

 outward from the expanding gas sphere boundary. 



In the later stages of the motion, the pressure in the gas sphere and 

 surrounding fluid falls below the hydrostatic value Po, the outward flow 

 is brought to rest, and then inward flow begins. The kinetic energy of 

 this motion is thus returned to compression of the gas sphere, rather 

 than being radiated to infinity as a wave of compression. At points 

 l^ehind the shock front, for which the pressures are large and rapidly 

 changing, the particle velocity depends on both the past history of the 

 pressure and its value at the time, and a clear cut distinction between 

 motion resulting directly from compression and noncompressive flow 

 cannot be made in this region. The natural attempt to discuss the two 

 types of motion as distinct must therefore run into difficulties, and the 

 fact that the two approximations are not mutually exclusive under these 

 conditions must be remembered. 



2.3. Waves of Finite Amplitude 



In the derivation of solutions for waves of small amplitude, a num- 

 ber of simplifying assumptions were made which led to well-known 

 forms of the wave equation appropriate to the type of motion assumed 

 possible. Although the approximations are amply justified for the 

 small variations of density and pressure developed by sound sources in 

 air or underwater, we have no reason to suppose that they could be for 

 the conditions existing in the gaseous products of an explosion or in the 

 water in the near vicinity. Before considering solutions of the exact 

 equations, the differences which must result can profitably be considered 

 by more qualitative arguments. 



It has been assumed that the quantity c = WdP/dp entering the 

 equations could be treated as a constant Co independent of the pressure 

 or state of motion in the fluid, and that the velocities of the fluid were 

 always negligible. These assumptions led to the result that any part 

 of a wave disturbance is propagated with a velocity c relative to a fixed 

 system of coordinates. If, however, this small amplitude velocity of 

 propagation c depends on the density and, as we should expect quali- 

 tatively, this velocity at any point is properly measured with respect to 

 coordinates moving with the liquid at the point, it is easy to see that 

 matters become more complicated. Consider first the variations of c 

 with density to be expected in real liquids and gases. The relation 

 P = P{p) is, of course, simply a curve of adiabatic compression and is, 

 for all normal fluids, concave upward, with the result that the slope 

 dP/dp and c = \^dP/dp increase with increasing compression. The 



