247 
Sec. 18 -36- 
When the rarefaction catches up, reflection will presumably ensue. 
At this point the simple Riemann theory will no longer apply because when 
the shock wave amplitude is reduced the medium back of the reduced shock 
wave is not on the same adiabatic as the medium through which the un- 
diminished shock wave passed, 
Although it may be quite difficult to calculate the exact law of 
decay, it is clear that the shock wave must decay and that the way in which 
it will decay will depend on the time during which the piston is moving and 
the way in which the piston is decelerated. This problem is under con- 
Sideration, 
Incidentally, it is at least conceivable that more effect could be 
produced at a long distance by a gradual acceleration of the piston than 
by @ sudden acceleration. During the initiation period the processes 
occurring are reversible until a discontinuity is produced. By a slow ac- 
celeration the distance in front of the piston at which discontinuity occurs 
is increased and therefore the degradation of energy is postponed. 
There have been many criticisms of the Huconiot treatment (Lamb,*? 
Rayleigh)}*the difficulty being connected with the idea that the passage of 
the shock front is an irreversible process although there is no viscosity or 
thermal conduction. As Rayleigh points out, perhaps one should consider the 
Hugoniot equations as limiting equations for very small viscosity, etc. Then 
the existence of a very steep shock front could result in a finite dissipa- 
tion of energy even though the effect of viscosity, etc., could be neglected 
elsewhere. 
18. Rarefaction Wave Following a Detonation Wave. 
In this section detonation waves initiated by a single impulse from 
a piston will be discussed. At the discontinuous detonation frontthe 
Hugoniot conditions must again be satisfied, with the modification that the 
change in chemical energy due to the reaction must appear in AE. In the 
shock wave the velocity (Uj) of the gas back of the wave was that of the 
piston and ali the energy came from the piston. Now, however, the détona- 
tion itself supplies energy and there is no reason why Up should equal w. 
There i8 another condition which must be satisfied at the detonation 
front, a condition which was discussed in Sec, 5. Upon the introduction of 
this condition, Po, Vo, and Up become fixed without reference to the piston 
speed, so that in order to keep the product gases at a constant velocity, 
the piston velocity w must be Specified, instead of being one of the 
independent variables as in the shock wave case. 
Therefore if it is assunicd that the detonation reaction starts 
instantly, the situation is quite simple when the piston is instantly 
accelerated to the proper final velocity w. Fis, 18-1 shows this case. 
we 
