3.12.5 Sound Speed Ratio for Weak Shocks
The sound speed ratio is given by the equation
c 2
c 1
¼
ffiffiffiffiffi
T 2
T 1
r
and by using Eq. (3.60) for the temperature ratio one can show, after some algebra,
that the sound speed ratio is given by
c 2
c 1
¼ 1 þ
γ À 1
2
u
c 1
þ
γ À 1
ð
Þ γ þ 1
2
À
Á
64
u
3
c
3
1
:
ð3:61Þ
Note that the term of order u
2
=c
2
1 drops out, hence, we deduce that the sound
speed ratio in the case a weak shock and that of a simple wave (see, for example,
Eq. (2.18)) agree up to terms of second order in the shock strength.
3.12.6 Entropy Change for Weak Shocks
Recalling our discussion in Sect. 1.6 of Chap. 1 and in Sects. 3.7 and 3.9 of Chap. 3,
we found that the entropy change (s 2 À s 1 ) per unit mass of fluid crossing the shock
can be written in the form,
s 2 À s 1
c V
¼ ln
p 2
p 1
À γ ln
ρ 2
ρ 1
:
Using previous expressions for the ratios, p 2 /p 1 and ρ 2 /ρ 1 , we have
s 2 À s 1
c V
¼ ln 1 þ γ
u
c 1
þ γ
γ þ 1
4
u
2
c 2
1
þ γ
γ þ 1
ð
Þ
2
32
u
3
c
3
1
!
À γ ln 1 þ
u
c 1
þ
3 À γ
4
u
2
c 2
1
þ
γ
2
À 14γ þ 17
ð
Þ
32
u
3
c
3
1
!
Using the expansion, ln 1 þ x
ð
Þ¼x À
x
2
2 þ
x
3
3 À . . . for terms up to third order for
small values of x, we can write the expression for the entropy change as
s 2 À s 1
c V
¼ γ
u
c 1
þ γ
γ þ 1
4
u
2
c 2
1
þ γ
γ þ 1
ð
Þ
2
32
u
3
c 3
1
&
'
À
1
2
γ
2 u
2
c 2
1
þ 2γ
2 γ þ 1
4
u
3
c 3
1
&
'
þ
γ
3
3
u
3
c 3
1
À γ
u
c 1
þ
3 À γ
4
u
2
c 2
1
þ
γ
2
À 14γ þ 17
ð
Þ
32
u
3
c
3
1
&
'
À
1
2
u
2
c 2
1
þ 2
3 À γ
4
u
3
c
3
1
&
'
þ
1
3
u
3
c
3
1
!
3.12 Approximate Analytical Expressions for Weak Shock Waves
121
The sound speed ratio is given by the equation
c 2
c 1
¼
ffiffiffiffiffi
T 2
T 1
r
and by using Eq. (3.60) for the temperature ratio one can show, after some algebra,
that the sound speed ratio is given by
c 2
c 1
¼ 1 þ
γ À 1
2
u
c 1
þ
γ À 1
ð
Þ γ þ 1
2
À
Á
64
u
3
c
3
1
:
ð3:61Þ
Note that the term of order u
2
=c
2
1 drops out, hence, we deduce that the sound
speed ratio in the case a weak shock and that of a simple wave (see, for example,
Eq. (2.18)) agree up to terms of second order in the shock strength.
3.12.6 Entropy Change for Weak Shocks
Recalling our discussion in Sect. 1.6 of Chap. 1 and in Sects. 3.7 and 3.9 of Chap. 3,
we found that the entropy change (s 2 À s 1 ) per unit mass of fluid crossing the shock
can be written in the form,
s 2 À s 1
c V
¼ ln
p 2
p 1
À γ ln
ρ 2
ρ 1
:
Using previous expressions for the ratios, p 2 /p 1 and ρ 2 /ρ 1 , we have
s 2 À s 1
c V
¼ ln 1 þ γ
u
c 1
þ γ
γ þ 1
4
u
2
c 2
1
þ γ
γ þ 1
ð
Þ
2
32
u
3
c
3
1
!
À γ ln 1 þ
u
c 1
þ
3 À γ
4
u
2
c 2
1
þ
γ
2
À 14γ þ 17
ð
Þ
32
u
3
c
3
1
!
Using the expansion, ln 1 þ x
ð
Þ¼x À
x
2
2 þ
x
3
3 À . . . for terms up to third order for
small values of x, we can write the expression for the entropy change as
s 2 À s 1
c V
¼ γ
u
c 1
þ γ
γ þ 1
4
u
2
c 2
1
þ γ
γ þ 1
ð
Þ
2
32
u
3
c 3
1
&
'
À
1
2
γ
2 u
2
c 2
1
þ 2γ
2 γ þ 1
4
u
3
c 3
1
&
'
þ
γ
3
3
u
3
c 3
1
À γ
u
c 1
þ
3 À γ
4
u
2
c 2
1
þ
γ
2
À 14γ þ 17
ð
Þ
32
u
3
c
3
1
&
'
À
1
2
u
2
c 2
1
þ 2
3 À γ
4
u
3
c
3
1
&
'
þ
1
3
u
3
c
3
1
!
3.12 Approximate Analytical Expressions for Weak Shock Waves
121