and in terms of the slopes of these characteristics as observed in a tx-diagram we
write these latter equations in inverted form as
dt
dx
C þ
¼
5
3R þ þ 2R À
and
dt
dx
C À
¼
5
2R þ þ 3R À
:
The head of the expansion fan has u ¼ 0 and c ¼ c 0 ¼ 1.2, hence, R + ¼ 6 and the
tail of the expansion fan has u ¼ À 0.75 and c ¼ c 0 À
γÀ1
2
À Á
0:75, hence, R + ¼ 4.5.
Let us now divide the expansion fan into seven
1 positive characteristics ranging from
R + ¼ 4.5 to R + ¼ 6 as illustrated in Fig. A.6. The negative characteristic ag has
R À ¼ À 6 and we have already determined that the equation for this characteristic is
given by
x t
ð Þ ¼ À
2c 0 t
γ À 1
þ
γ þ 1
γ À 1
c 0 t 0
t
t 0
3Àγ
1þγ
,
where t 0 ¼ L/c 0 ¼ 10/1.2 ¼ 8.333, so that the coordinates (x, t) of position a as
shown in Fig. A.6 is (10, 8.333).
Fig. A.6 Intersection points
(a, b, c. . . g) of the positive
characteristics in the
expansion fan with the
negative characteristic ag,
and two of the points, P 1 and
P 2 , in the region of interest
are displayed
1 Greater accuracy is achieved with a larger number of characteristics but at the cost of an increased
number of points to be computed in region ℜ 4 .
Appendix A
323