HYDRODYNAMICAL RELATIONS 65 



density. The rarefaction wave is thus the negative of the initial shock 

 wave but is propagated into a fluid having a velocity u toward its front 

 and the boundary. The rarefaction must change this velocity by an 

 amount —u relative to the direction of its advance away from the 

 boundary, and hence leave the fluid with a velocity 2u toward the 

 boundary. Relative to the undisturbed fluid, the front thus has a 

 velocity U — 2u away from the boundary, U being the velocity of the 

 incident plane wave front, and the boundary is displaced away from 

 the fluid with velocity 2u. 



The velocity doubling for acoustic waves is thus found for waves of 

 finite amplitude at normal incidence because of the apparent reversi- 

 bility of the Rankine-Hugoniot conditions w^hich involve only relative 

 velocities. The only differences are that the velocity u is related to 

 the incident pressure by the Rankine-Hugoniot conditions, and the 

 front of the rarefaction wave travels with a decreased velocity. The 

 particle velocity in terms of pressure is given from Eq. (2.28) by 



P — P 

 (2.42) u = ^—f^ 



poU 



and this differs only from the acoustic case by use of the shock front 

 velocity U instead of the sound velocity Co. The velocity U — 2u oi 

 the rarefaction front is for water very nearly equal to the acoustic 

 velocity Co. 



The derivation of Eq. (2.42) assumes the reversibihty of the shock 

 front process. This is of course not defensible, as energy is degraded 

 into heat by the passage of a compression wave. The fluid behind the 

 front is therefore hotter and cannot be returned to its initial pressure 

 without a new" value of density. A second associated difficulty is the 

 fact that, even if a discontinuous rarefaction front is developed at the 

 boundary, the a.dvance of this front must involve a progressive decrease 

 in the pressure gradient, as discussed in section 2.3. In view of these 

 difficulties, it is perhaps surprising that steep rarefaction fronts are in 

 fact observed, and that the velocity doubling for normal incidence has 

 been confirmed by experiment at pressures up to 20,000 Ib./in.^ (see 

 section 10.2). It should equally be no surprise if the simple rarefaction 

 scheme is found not to apply at higher pressures or oblique incidence. 



The phenomena at oblique incidence have not been examined in 

 any work reported at the present time. It is evident, however, that 

 some readjustment must occur to take into account the acoustic para- 

 dox for grazing incidence on a boundary, at which the incident wave 

 produces only tangential velocity changes and yet the fluid must be 

 reduced to its initial pressure. This sort of complication must occur 

 even in the idealization of a plane wave of infinite duration and negli- 



