241 
Sec. 14 -30- Eq. 99-105 
For a perfect cas P = k eC ‘ a“P/a ies 2 we, sein (3-2 ; 
eee AY, ape eo (99) 
y [2 2 C (a°P/a - 2) /ace | 
Presumably this result will still be true for imperfect gases. We do not 
kmow what the result will be for water. A similar result is obtained for s 
lines, 
7. Only in exceptional circumstances (U + c = velocity of piston) can 
S or r lines run parallel to the piston curves (in x, t plane). Ordinarily 
these lines will end on the piston curves. At these ends,U = U of piston, 
Therefore if s (or r) is known for a line, U,W ,(, c and P are determined 
by U of the piston at the point where the line ends or begins at a piston. 
14. Solution for Progressive Waves. 
In certain cases s = 3 (W - U) is constant over a region. The r lines 
are then straight as already mentioned. Furthermore the solutions of 
Riemann's equations are readily obtained for such a region. For if 
Se 4 (WJ = U) = const,, (100) 
then 6) = 28+ U =U + const., (101) 
so that Zq. (95) becomes 
U ou 
een WU el ee ‘ (102) 
The solution of this equation (Rayleich)is 
Do= f(x (eee (103 ) 
where f is an arbitrary function. The proof of this is 
ou ia dc, a 
eS tices dc U af 
re =| t(l+ ay) 52 a8 A 
ati e'/\1+ te" (1+ 92) (104 ) 
dx au * : 
where f is the derivative of f. Also 
ou [' dc ou] ' 
aes = 10) iG = fy } 
ae (U4 0) + (l+ ay) Se | : 
Luge (y+ c)e'/ [2 phere (ue $2) | (105) 
4 oer aU , 
from which Eq. (102) follows. 
