Air is a gas, and a very important
property
of any gas is the speed of sound through the gas. Why
are we interested in the speed of sound? The speed of "sound"
is actually the speed of transmission of a small disturbance through
a medium. Sound itself is a sensation created in the human
brain in response to sensory inputs from the inner ear.
(We won't comment on the old
"tree falling in a forest" discussion!)
Disturbances are transmitted through a gas as a result of
collisions
between the randomly moving molecules in the gas.
The transmission of a small disturbance through a gas is an
isentropic process. The conditions in the
gas are the same before and after the disturbance passes through.
Because the speed of transmission depends on molecular collisions,
the speed of sound depends on the state
of the gas. The speed of sound is a constant within a given gas
and the value of the constant depends on the type of gas (air, pure oxygen,
carbon dioxide, etc.) and the temperature of the gas. For
hypersonic flows,
the high temperature of the gas generates
real gas effects
that can alter the speed of sound. An
analysis
based on conservation of
mass
and
momentum
shows that the square of the speed of sound a^2 is equal to the
the gas constant R times the
temperature T
times the ratio of.
specific heatsgamma
a^2 = R * T * gamma
Notice that the
temperature
must be specified on an absolute scale (Kelvin
or Rankine). The dependence on the type of gas is included in the
gas constant R. which equals the universal gas constant divided by the
molecular weight of the gas, and on the ratio of specific heats.
If the
specific heat capacity
of a gas is a constant value, the gas is said
to be calorically perfect and if the specific heat capacity
changes, the gas is said to be calorically imperfect.
At subsonic and low supersonic
Mach numbers, air is calorically perfect.
But under low hypersonic conditions, air is
calorically imperfect.
Derived flow variables, like the speed of sound and the
isentropic flow relations
are slightly different for a calorically imperfect gas
than the conditions predicted for a calorically perfect gas
because some of the energy of the flow excites the vibrational
modes of the diatomic molecules of nitrogen and oxygen in the air.
The equation given above was derived for a calorically perfect gas.
If we include the effects of caloric imperfection, some additional
terms are added to the equation.
For the calorically imperfect case,
mathematical models
based on a simple harmonic vibrator have been developed.
The details of the analysis were given by Eggars in
NACA Report 959.
A synopsis of the report is included in
NACA Report 1135.
To a first order approximation, the equation for the speed of sound for
a calorically imperfect gas is given by:
where theta is a thermal constant
equal to 3056 degrees Kelvin, gamma is the calorically perfect
value of the ratio of specific heats, and T is the static temperature.
The speed of sound in air depends on the type of gas and
the temperature of the gas. On
Earth, the atmosphere is composed of
mostly diatomic nitrogen and oxygen, and the temperature
depends on the altitude in a rather complex way.
Scientists and engineers have created a
mathematical model of the atmosphere to help
them account for the changing effects of temperature with altitude.
We have created an
atmospheric calculator
to let you study the variation of sound speed with altitude.
Here's another Java program to calculate speed of sound and
Mach number
for different altitudes and speed. You can use this calculator
to determine the Mach number of an aircraft at a given speed and altitude.
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 English or Metric units and you can input either the Mach number
or the speed by using the menu buttons. Just click on the menu button and click
on your
selection. There is a
sleek version
of this program for experienced users who do not need these instructions.
You can also download your own copy of this program to run off-line by clicking
on this button:
