Finally, tidying up and substituting back the values for A and B, we obtain the
following equation for the family of negative characteristics,
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 :
ð2:81Þ
We can obtain expressions for the pressure, density and temperature in the
expansion fan by noting that isentropic conditions apply throughout. The isentropic
relation,
p
ρ γ ¼ constant
and this taken in conjunction with the equations, p ¼ ρRT and c /
ffiffiffi ffi
T
p
, it follows
that
T
T 0
¼
c
c 0
2
,
p
p 0
¼
c
c 0
2γ
γÀ1
,
ρ
ρ 0
¼
c
c 0
2
γÀ1 :
Taking Eq. (2.75) and dividing by c 0 gives
c
c 0
¼
γ À 1
γ þ 1
x
c 0 t
þ
2
γ þ 1
,
hence,
T
T 0
¼
γ À 1
γ þ 1
x
c 0 t
þ
2
γ þ 1
! 2
,
ð2:82Þ
p
p 0
¼
γ À 1
γ þ 1
x
c 0 t
þ
2
γ þ 1
! 2γ
γÀ1 :
ð2:83Þ
ρ
ρ 0
¼
γ À 1
γ þ 1
x
c 0 t
þ
2
γ þ 1
! 2
γÀ1 :
ð2:84Þ
(see Appendix A for more details)
80
2 Waves of Finite Amplitude