1
2
ρv
3
þ eρv,
and, in general, the work (pressure  area  distance) done per unit area per unit
time by the pressure forces is
pv:
Applying these equations to the energy conservation condition as expressed
above gives
1
2
ρ U s À u p
À
Á 3 þ ρe U s À u p
À
Á
h
i
À
1
2
ρ 0 U
3
s þ ρ 0 e 0 U s
h
i
¼ p 0 U s À p U s À u p
À
Á
,
where e and e 0 are the internal energies per unit mass leaving and entering the shock
front, respectively. The latter equation can be written 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 3 þ ρe U s À u p
À
Á þ p U s À u p
À
Á ð3:3Þ
Let us now spend some time manipulating these equations in order to establish
some useful relationships that can be used at a later stage. Eq. (3.1) gives
u p ¼ 1 À
ρ 0
ρ
U s ,
ð3:4Þ
and if we write Eq. (3.2) as
ρ 0 U
2
s À ρ U s À u p
À
Á
U s À u p
À
Á ¼ p À p 0
and substituting Eq. (3.1) in this latter equation we have
ρ 0 U
2
s À ρ 0 U s U s À u p
À
Á ¼ p À p 0 ,
hence,
ρ 0 U s u p ¼ p À p 0 :
ð3:5Þ
Multiplying Eq. (3.5) by Eq. (3.4) gives
ρ 0 U s u
2
p ¼ p À p 0
ð
Þ 1 À
ρ 0
ρ
U s ,
so that the fluid velocity is
3.2 Conservation Equations
91