dx
dt
¼
u 0
2
γ À 3
ð
ÞÀc 0
h
i
and substituting the values for u 0 , γ and c 0 in this latter equation, we find that (dx/
dt) ¼ À 1.903. However, this value for the slope should correspond to the slope of
the negative characteristic (P-A) at point A in Fig. A.2; differentiating this negative
characteristic, we find that
dx
dt
¼ À
2c 0
γ À 1
þ
3 À γ
γ À 1
c 0
t
t 0
2 1Àγ
ð
Þ
1þγ
,
and by using this latter equation we find the slope at t ¼ 1.641 as,
dx
dt
t¼1:641
¼ À1:904
(where t 0 ¼ 1) and this is, therefore, in agreement with the slope of the negative
characteristic in ℜ3. By using the intersection values above, we can write the
equation for the negative characteristic (A-E) in ℜ3 as
x t
ð Þ ¼
u 0
2
γ À 3
ð
ÞÀc 0
h
i
t À 1:641
ð
Þþ0:169
and this characteristic intersects the piston’s path at E in Fig. A.2 and it is easy to
verify that this intersection occurs at t ¼ 3.281 and, accordingly, this negative
characteristic is shown plotted in the range: 1.641 t 3.281.
Particle-Path
The particle velocity u within the expansion fan has the following dependence on
x and t according to Eq. (2.74),
u ¼
2
γ þ 1
x
t
À c 0
h
i
:
In order to determine the path taken by particles we must solve the equation,
dx
dt
¼
2
γ þ 1
x
t
À c 0
h
i
with the initial particle position at x ¼ c 0 t 0 when the particle begins to move. By
carrying out a similar integration to that performed for the negative characteristic as
316
Appendix A
dt
¼
u 0
2
γ À 3
ð
ÞÀc 0
h
i
and substituting the values for u 0 , γ and c 0 in this latter equation, we find that (dx/
dt) ¼ À 1.903. However, this value for the slope should correspond to the slope of
the negative characteristic (P-A) at point A in Fig. A.2; differentiating this negative
characteristic, we find that
dx
dt
¼ À
2c 0
γ À 1
þ
3 À γ
γ À 1
c 0
t
t 0
2 1Àγ
ð
Þ
1þγ
,
and by using this latter equation we find the slope at t ¼ 1.641 as,
dx
dt
t¼1:641
¼ À1:904
(where t 0 ¼ 1) and this is, therefore, in agreement with the slope of the negative
characteristic in ℜ3. By using the intersection values above, we can write the
equation for the negative characteristic (A-E) in ℜ3 as
x t
ð Þ ¼
u 0
2
γ À 3
ð
ÞÀc 0
h
i
t À 1:641
ð
Þþ0:169
and this characteristic intersects the piston’s path at E in Fig. A.2 and it is easy to
verify that this intersection occurs at t ¼ 3.281 and, accordingly, this negative
characteristic is shown plotted in the range: 1.641 t 3.281.
Particle-Path
The particle velocity u within the expansion fan has the following dependence on
x and t according to Eq. (2.74),
u ¼
2
γ þ 1
x
t
À c 0
h
i
:
In order to determine the path taken by particles we must solve the equation,
dx
dt
¼
2
γ þ 1
x
t
À c 0
h
i
with the initial particle position at x ¼ c 0 t 0 when the particle begins to move. By
carrying out a similar integration to that performed for the negative characteristic as
316
Appendix A