u p ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p À p 0
ð
Þ
1
ρ 0
À
1
ρ
s
:
ð3:6Þ
Similarly, if Eq. (3.4) is used in the latter equation we obtain the shock velocity in
terms of pressure and density according to
U s ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ρ
ρ 0
p À p 0
ρ À ρ 0
s
:
ð3:7Þ
By writing Eq. (3.3) in the form;
1
2
ρ 0 U
3
s þ ρ 0 e 0 U s þ p 0 U s ¼
1
2
ρ U s À u p
À
Á 2 U s À u p
À
Á þ ρe U s À u p
À
Á þ p U s À u p
À
Á
and using Eq. (3.1) we have
1
2
ρ 0 U
3
s þ ρ 0 e 0 U s þ p 0 U s ¼
1
2
ρ 0 U s U s À u p
À
Á 2 þ ρ 0 U s e þ
pρ 0
ρ
U s :
Cancelling ρ 0 U s across gives
1
2
U
2
s þ e 0 þ
p 0
ρ 0
¼
1
2
U s À u p
À
Á 2 þ e þ
p
ρ
,
ð3:8Þ
and re-arranging this latter equation we obtain
e À e 0 ¼
p 0
ρ 0
À
p
ρ
þ U s u p À
1
2
u
2
p :
Substituting Eqs. (3.5) and (3.6) in this latter equation gives
e À e 0 ¼
p 0
ρ 0
À
p
ρ
þ
p
ρ 0
À
p 0
ρ 0
À
1
2
p À p 0
ð
Þ
1
ρ 0
À
1
ρ
so that the increase in internal energy as a result of the shock is
e À e 0 ¼
1
2
p þ p 0
ð
Þ
1
ρ 0
À
1
ρ
,
ð3:9Þ
which is called the Hugoniot equation. Other equations based on the conservation
laws are known as the Rankine-Hugoniot equations and they will be considered in
due course.
92
3 Conditions Across the Shock: The Rankine-Hugoniot Equations