x t
ð Þ ¼ Àu 0 t 0 þ c 0 À
γ þ 1
2
u 0
h
i
t À t 0
ð
Þ
and with t 0 ¼ 1 (at position D) this equation becomes
x t
ð Þ ¼ Àu 0 þ c 0 À
γ þ 1
2
u 0
h
i
t À 1
ð
Þ
¼ 0:103t À 1:003; t ! 1
Parameter Variations According to the Method
of Characteristics
The particle velocity, the pressure and the density within the expansion fan are given
by the equations (see Sect. 2.7.2);
u ¼
2
γ þ 1
x
t
À c 0
h
i
,
p
p 0
¼
γ À 1
γ þ 1
x
c 0 t
þ
2
γ þ 1
! 2γ
γÀ1
and
ρ
ρ 0
¼
γ À 1
γ þ 1
x
c 0 t
þ
2
γ þ 1
! 2
γÀ1
,
respectively, in the range; c 0 À
γþ1
2
À Á
u 0
Â
Ã
t x c 0 t. In the specific case considered
here, namely, at time t ¼ t 0 ¼ 1, p 0 ¼ 1, ρ 0 ¼ 1 and c 0 ¼ 1.183, we have,
u x
ð Þ ¼
2
γ þ 1
x À 1:183
½
Š
p x
ð Þ ¼
γ À 1
γ þ 1
x
1:183
þ
2
γ þ 1
! 2γ
γÀ1
and
ρ x
ð Þ ¼
γ À 1
γ þ 1
x
1:183
þ
2
γ þ 1
! 2
γÀ1
318
Appendix A