As an object moves through a gas, the gas molecules are deflected
around the object. If the speed of the object 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 object 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. Shock waves
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.

Due to IT
security concerns, many users are currently experiencing problems running NASA Glenn
educational applets. The applets are slowly being updated, but it is a lengthy process.
If you are familiar with Java Runtime Environments (JRE), you may want to try downloading
the applet and running it on an Integrated Development Environment (IDE) such as Netbeans or Eclipse.
The following are tutorials for running Java applets on either IDE:
Netbeans Eclipse

To change input values, click on the input box (black on white),
backspace over the input value, type in your new value. Then hit
the red COMPUTE button to
send your new value to the program.
You will see the output boxes (yellow on black)
change value. You can use either Imperial or Metric units and you can set default
conditions for the Earth or Mars.
Just click on the menu button and click on your selection.
For the given altitude input, the program computes the upstream
standard day conditions. The ratios are computed
using the
Mach number
and
specific heat ratio
inputs and the equations given in the figure. The
downstream conditions are computed from the upstream conditions and
the ratios.
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:

There is more complete
shock simulation
program that is also available at this web site. The
ShockModeler
program models the intersection and reflection of multiple shock waves.