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
in the flow.
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.
Near and beyond
the speed of sound (about 330 m/s or 700
mph on Earth at sea level),
small disturbances in the flow are transmitted
to other locations
isentropically (with constant entropy)
as sound waves.

For
supersonic and hypersonic
flows, small disturbances are transmitted
downstream within a cone.
The edge of the cone is depicted two-dimensionally by the blue lines
on the figure at the top of this page.
The sound waves strike the edge of the cone at a right angle and the
speed of the sound wave is denoted by the letter a.
The flow is moving at velocity v which is greater than a.
From
trigonometry, the
sine
of the cone angle mu is
equal to the ratio of a and v:

sin(mu) = a / v

But the ratio of v to a is the
Mach number
of the flow.

M = v / a

With a little algebra, we can determine that the cone angle mu
is equal to the
inverse sin of one over the Mach number.

sin(mu) = 1 / M

mu = asin(1 / M)

where asin is the trigonometric
inverse sine function.
It is also written as shown
on the slide sin^-1.
Mu is an angle which depends only on the Mach number
and is therefore called the
Mach angle of the flow.

We are interested in determining the Mach angle because small
disturbances in a supersonic flow are confined to the cone formed
by the Mach angle.
There is no upstream influence in a supersonic flow; disturbances
are only transmitted downstream within the cone.

Here's a JavaScript program which solves for the Mach angle .

Isentropic Flow Calculator

Input

Output

Mach

Mach Angle

P-M Ang

p/pt

T/Tt

rho/rhot

q/p

A/A*

Wcor/A

You select an input variable by using the choice button labeled Input
Variable. Next to the selection, you then type in the value
of the selected variable. When you hit the red COMPUTE button,
the output values change.
The default input variable is the Mach number, and by varying Mach number
you can see the effect on Mach angle. You can also select
Mach angle as an input, and see its effect on the other flow variables.

If you are an experienced user of this calculator, 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 the yellow button.
Look for the Isentropic Flow Calculator.