Substituting Eqs. (3.25) and (3.28) in this latter equation yields
ΔS
mc V
¼ ln
1 À γ þ 2γM
2
1
γ þ 1
2 þ γ À 1
ð
ÞM
2
1
γ þ 1
ð
ÞM
2
1
! γ
,
ð3:33Þ
where ΔS ¼ S 2 À S 1 and Eq. (3.33) is shown plotted in Fig. 3.5. Defining,
ΔS
mc V
ln f x
ð Þ
½
Š,
where
f x
ð Þ ¼
1 À γ þ 2γx
γ þ 1
2 þ γ À 1
ð
Þx
γ þ 1
ð
Þx
! γ
ð3:34Þ
and x M
2
1 . It can be seen that f(1) ¼ 1, so that S 2 À S 1 ¼ 0, hence, there is no
entropy increase when the upstream Mach number is unity and we have noted that
M 2 ¼ 1 when M 1 ¼ 1, so there is no shock and no discontinuity.
Fig. 3.5 The change in entropy across a normal shock is shown (γ ¼ 1.4)
3.9 Entropy Change Across the Shock in Terms of Mach Number
107