242 
Sec. 15 -31- 
This solution implies that c is a function of U alone. That this is 
correct follows from the fact that P and therefore c is a function of ie 
only and that @ is a function of U through Iq. (90) and (101). However, c 
will be a different function of U for different (constant) values of 8. 
15. Simple Rarefaction Waves, 
Graphical method. One of the simplest applications is to the 
case er an infinite tube, closed at one end by a movable piston. Everything 
is initially at rest and. at time t = ty the piston begins to move backwards, 
thus initiating a rarefaction wave which travels down the tube to the right, 
away from the piston. Fig. 15-1 shows the path of the piston and the 8 and. r 
lines, 
{ / 
Cae } 
| Hai tik 
| \ 
Fig. 15-1 
t 
Since at t = 0 the fluid is assumed to be uniform and at rest, U and@= 0 
at t = 0( Conte taken as initial density). Therefore s and r = O in the 
4 
region I so that both sets of lines are straic ght. The s lines from region 
I cover the whole x - t plane, since the tube is infinitely long. There- 
fore 8 = O everywhere and the r lines are straight everywhere and are 
really contour lines for U. The value of U for any r line equals the 
value of U of the piston at the starting point of the line. The lines 
Starting before t = t, thus have U= 0. Between t, and tp the piston has 
a negative acceleration so the r lines slope less and less steeply and 
correspond to values of U decreasing from U = 0 to U = - w (-w is final 
velocity of piston, attained at t = to). Since r is changing from line to 
line in region II, the s lines are curved as shown (dU <0). In region ITI, 
U is again constant (=.w) so that both sets of lines are straight in this 
region, 
b. Analytical method. The analytic solution is obtained as follows, 
Let X and 7 represent x and t for the piston. Then the coordinates x, t of 
an r line of value U starting from the piston point X,7 are related by the 
