243 
Sec. 16 -32- Eq. 106 
equation 
Kok + (tr=-7 jul wio)}e. (106) 
Let X = X(7 ), then U = axX/d7 = X'(7’) at time 7”. Also c is a mown 
function of U, therefore of ? . Consequently, Eq. (106) can be converted 
into an equation involving only x, 7 and t, or by solving U = ax/a7 = x 
(7) for 7 in terms of U, an equation involving only x, t and U can be ob- 
tained. Solution of this for U gives U as a function of x and t, and 
therefore also @, P and c as a function of x and t, the complete solution 
of the problem,’ Rayleich gives some explicit results when the acceleration 
of the piston is constant from t, to to. 
It will be noted that the front of the rarefaction wave propagates 
with the velocity of sound in the original medium, while the back propagates 
with the velocity of sound in the final rarefied medium plus the (negative ) 
mass velocity of the final medium. The rarefaction wave therefore broadens 
out as it passes down the tube, and no discontinuities are produced. 
16. Simple Shock Waves in Air and in Water. 
If the piston is instantaneously accelerated to its final velocity w, 
the problem is readily soluble, as was shown in Sec. 4a, where the fluid 
was assumed to be a perfect gas with constant heat capacity. 
a. Air, variable heat capacity. The heat capacity of air actually 
varies with temperature, Bethe@and Teller? have carefully investigated 
Shock waves in air, taking this into account. Their results are given 
below in Table 16-1. 
These investigators also studied very carefully the effect of the 
finite time required for translation, rotation, vibration and dissociation 
to come to equilibrium. These lags, especially that of dissociation, alter 
the shape of the pressure and temperature rise, In Table 16-1, there are 
two sets of entries. The first is calculated on the assumption that all 
degrees of freedom, including dissociation into free atoms, are in 
equilibrium. The second set is calculated on the assumption that there is 
no time for dissociation or vibration to change. Comparing the two cases, 
one sees that if equilibrium is established, the shock velocity D is less; 
the density higher, the temperature lower, and the pressure less than in 
the case when equilibrium is not completely established. Presumably the 
equilibrium values are the proper cones to use unless there is a rarefaction 
wave or some other phenomenon immediately behind the shock front. 
At the present time Table 16-1 is being extended to higher velocities. 
b. Shock waves in water. Shock waves in water differ greatly from 
those in air. If the pressure difference is the same, the shock velocity 
and the mass velocity are considerably less, and the temperature rise is 
enormously less. If the piston velocity (equals mass velocity) is the same 
