As a rocket moves through a gas, the gas molecules are deflected
around the rocket. If the speed of the rocket is much less than the
speed of sound
of the gas, the density of the gas remains constant and the flow of
gas can be described by conserving momentum and energy.
As the
speed of the rocket increases towards the speed of sound, we
must consider
compressibility effects
on the gas. The density of the gas varies locally as the gas is
compressed by the object.

For compressible flows with little or small
flow turning, the flow process is reversible and the
entropy
is constant.
The change in flow properties are then given by the
isentropic relations
(isentropic means "constant entropy").
But when an object moves faster than the speed of sound,
and there is an abrupt decrease in the flow area,
the flow process is irreversible and the entropy increases.
Shock waves are generated
which are very small regions in the gas where the
gas properties
change by a large amount.
Across a shock wave, the static
pressure,
temperature,
and gas
density
increases almost instantaneously.
Because a shock wave does no work, and there is no heat addition, the
total
enthalpy
and the total temperature are constant.
But because the flow is non-isentropic, the
total pressure downstream of the shock is always less than the total pressure
upstream of the shock; there is a loss of total pressure associated with
a shock wave.
The ratio of the total pressure is shown on the slide.
Because total pressure changes across the shock, we can not use the usual (incompressible) form of
Bernoulli's equation
across the shock.
The
Mach number
and speed of the flow also decrease across a shock wave.

If the
shock wave is perpendicular to the flow direction it is called a normal
shock. On this slide we have listed the equations which describe the change
in flow variables for flow across a normal shock.
The equations presented here were derived by considering the conservation of
mass,
momentum,
and
energy.
for a compressible gas while ignoring viscous effects.
The equations have been further specialized for a one-dimensional flow
without heat addition.

The equations can be applied to the
two dimensional flow past a wedge for the following combination of
free stream Mach number M and wedge angle a :

where gam is the
ratio of specific heats.
If the wedge angle is less than this detachment angle, an attached
oblique shock
occurs and the equations are slightly modified.

Across the normal shock wave
the Mach number decreases to a value specified as M1:

The right hand side of all these equations depend only on the free stream
Mach number. So knowing the Mach number,
we can determine all the conditions associated with
the normal shock.
The equations describing normal shocks
were published in a NACA report NACA-1135
in 1951.

Here's a Java program based on the normal shock equations.
You can use this simulator to study the flow past a wedge.

Input to the program can be made
using the sliders, or input boxes at the upper right. To
change the value of an input variable, simply move the slider. Or
click on the input box, select and replace the old value, and
hit Enter to send the new value to the program.
Output from the program is displayed
in output boxes at the lower right. The flow variables are presented as ratios
to free stream values. The graphic at the left shows the wedge (in red)
and the shock wave generated by the wedge as a line. The line is colored
blue for an oblique shock and magenta when the shock is a normal shock. The black
lines show the streamlines of the flow past the wedge. Notice that downstream
(to the right) of the shock wave, the lines are closer together than upstream.
This indicates an increase in the density of the flow.
If you are an experienced user of this simulator, you can use a
sleek version
of the program which loads faster on your computer and does not include these instructions.
You can also download your own copy of the program to run off-line by clicking on this button: