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 approaches 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,
shock waves are generated in the flow.
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.
The changes in the flow
properties are irreversible and the
entropy
of the entire system increases.
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 inclined to the flow direction it is called an oblique
shock. On this slide we have listed the equations which describe the change
in flow variables for flow past a two dimensional wedge. The same changes and
oblique shocks occur downstream of a
nozzle
if the expanded pressure is different from free stream conditions.
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 two-dimensional flow
(not three dimensional axisymmetric) without heat addition.
The equations only apply for
those combinations of free stream Mach number and wedge angle for which
an attached oblique shock occurs. If the Mach number is too low, or the
wedge angle too high, the
normal shock
equations should be used.
For the Mach number change across an oblique shock there are
two possible solutions; one supersonic and one subsonic. In nature, the
supersonic ("weak shock") solution occurs most often. However, under some
conditions the "strong shock", subsonic solution is possible.

For the problem given on the slide, a supersonic flow at Mach number M
approaches a wedge of angle a. A shock wave is generated which is
inclined at angle s. Then:

The right hand side of all these equations depend only on the free stream
Mach number and the shock angle. The shock angle depends in a complex way on
the free stream Mach number and the wedge angle. So knowing the Mach number
and the wedge angle, we can determine all the conditions associated with
the oblique shock.
The equations describing oblique shocks
were published in NACA report
(NACA-1135)
in 1951.

Here's a Java program based on the oblique 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 lower left. 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 top 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.
There is more complete shock simulation program that is also avaliable at this web site. The
ShockModeler
program models the intersection and reflection of multiple shock waves.
You can also download your own copy of the program to run off-line by clicking on this button: