240 
Sec. 15 -29- Eq. 97-98 
These signify that the quantity «+ U appears to be constant to an observer 
moving; along the tube with a velocity U+ c while the quantity w - U appears 
constant to an observer moving with the velocity U - c. (These velocities 
are not necessarily constant. ) 
When applicable these equations enable one to see the nature of the 
solution. Two kinds of lines can be drawn in the x-t plane: "r lines", 
dx/at =U +c, along which 
r=3 (@ + U) is constant, 
and "s lines", dx/dt = U - c, along which 
s=s (WJ - U) is constant. 
These lines nave certain useful properties: 
1. There is a line of each type through each point of the x, t plane. 
2. If the values of s and r on the lines passing through a given 
point are known, then U and4J are known there. From these ( and P can be 
found from Itg. (90). 
3. If lines of a given kind having different values of r (or s) come 
together at a point, there will be a discontinuity in Dae at that point. 
4, If in a given region s has the same value along adjacent s lines, 
the r lines are straight in that region. Tor r is constant along an r line 
(which will cross the 3 lines) and if s is also constant along the r lines, 
both U andé) , therefore U and c, therefore the slope of the r lines must 
be constant. 
5) 
5. Likewise if r has the same value along adjacent r lines in a given 
region, the gs lines will be straight in this region. 
6. Fora perfect gas, at least, the curvature of an r or an 8 line is 
positive if U increases along the line. 
Proof; (r case) curvature is determined by 
a (U+ c) = AU+ (dc/aw jaw. (97) 
But cememes alr» {) (Ce ine 
Gli — lie Glen 1m fe ot 
and cf = aP/aP 0 2c(ac/aP ) = a?P/A PF? , 
so dc/aW = (a°P/a 2) /2c2 , @ positive quantity. But along an r line 
6& + U is constant so dW = -dU. Therefore 
d(U +c) = [a- C (a®P/a 0 2)/2c? | qt) ise (93) 
