and at t ¼ t 0 ¼ 1, x ¼ 1.183 (see Fig. A.2). The negative characteristic in region ℜ1
(see Fig. A.2) that passes through the point (x, t) ¼ (1.183, 1) is given by
x ¼ Àc 0 t À 1
ð
Þþ1:183; 1 ! t ! 0:
The equations for the negative characteristic according to Eq. (2.81), after passing
through the point (t 0 , c 0 t 0 ) ¼ (1, 1.183) on the positive characteristic that separates
regions ℜ1 and ℜ2), is given by
x t
ð Þ ¼ À
2c 0
γ À 1
t þ
γ þ 1
γ À 1
c 0 t 0
t
t 0
3Àγ
1þγ
,
while the slope of the negative characteristic in ℜ3 is
Fig. A.1 Diagram showing constant piston motion creating an expansion wave and a sketch in the
x-t plane of the piston path and the expansion fan
314
Appendix A