3.4 Alternative Notation for the Conservation Equations
Across the Shock
It is convenient at this stage to introduce new variables to define the conditions on
either side of the shock front; on one side let us define;
U s ! U 1 , ρ 0 ! ρ 1 , p 0 ! p 1 and T 0 ! T 1
and on the other side of the shock front we define;
U s À u p
À
Á ! U 2 , ρ ! ρ 2 , p ! p 2 and T ! T 2 ,
so that
u p ¼ U 1 À U 2
ð3:11Þ
where U 1 is the fluid velocity entering the shock and U 2 is the fluid velocity leaving
the shock as seen by an observer that moves with the shock.
Accordingly, the conservation equations now become
ρ 1 U 1 ¼ ρ 2 U 2
ð3:12aÞ
ρ 1 U
2
1 þ p 1 ¼ ρ 2 U
2
2 þ p 2
ð3:12bÞ
1
2
U
2
1 þ c P T 1 ¼
1
2
U
2
2 þ c P T 2
ð3:12cÞ
3.5 A Very Weak Shock
In the case of a very weak shock let us follow the analysis in the article by Blum
[7]. Suppose velocity, temperature and pressure are known on either side of the
shock and if we let
ρ 1 U 1 ¼ ρ 2 U 2 K 1
ð3:12dÞ
ρ 1 U
2
1 þ p 1 ¼ ρ 2 U
2
2 þ p 2 K 2
ð3:12eÞ
94
3 Conditions Across the Shock: The Rankine-Hugoniot Equations