applies. The only solution is to have the positive characteristics emanating from the
origin so that they do not intersect the characteristics, x ¼ c o t and x ¼ [c 0 À ((γ + 1)/
2)u 0 ]t. Consequently, these characteristics are given by
x
t
¼ u þ c
and as c ¼ c 0 + [(γ À 1)/2]u from Eq. (2.67), we can solve for u and c, giving,
u ¼
2
γ þ 1
x
t
À c 0
h
i
ð2:74Þ
and
c ¼
γ À 1
γ þ 1
x
t
þ
2c 0
γ þ 1
:
ð2:75Þ
The region ℜ2 is called an expansion fan or centred expansion and the characteristics are sketched in Fig. 2.18.
An infinite number of straight line characteristics can be drawn starting at x ¼ c 0 t
which is the head of the expansion fan and ending at x ¼ [c 0 À ((γ + 1)/2)u 0 ]t which
is the tail of the expansion fan.
As regards the negative characteristics in this region, we note that they are
determined by the equation
dx
dt
¼ u À c:
Substituting for u and c from Eqs. (2.74) and (2.75) gives
Fig. 2.18 A piston withdrawn at constant speed generates an expansion fan with the positive
characteristics emanating from the origin
78
2 Waves of Finite Amplitude