
Speed of Sound Derivation

Glenn
Research
Center

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, as described below.
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.
On this page we will derive the relationship between the speed of
sound and the state of the gas.
We begin with the
conservation of mass
equation:
Eq. 1:
mdot = r * u * A
where mdot is the mass flow rate, r is the
density
of the gas, u is the gas velocity, and A is the flow
area. Similarly, the one dimensional
conservation of momentum
equation specifies:
Eq. 2:
dp = r * u * du
where dp is the differential change in
pressure
and du is the differential change in velocity.
Let us assume that the flow area and mass flow rate are constant,
and the particular velocity that we are going to determine is the
speed of sound a. Then:
Eq. 3:
r * u = r * a = (r + dr) * (a + du)
Eq. 4:
r * a = r * a + r * du + a * dr + dr * du
where dr is a differential change in density and
du is a differential change in velocity. The last term in
Eq. 4 is very small, so let us ignore it to obtain:
Eq. 5:
r * du =  a * dr
Now substitute Eq. 5 into Eq. 2:
Eq. 6:
dp = a^2 * dr
For sound waves, the variations are small and nearly reversible.
We can then evaluate the change in pressure from the
isentropic relations.
Eq. 7:
dp / p = gamma * dr / r
where gamma is the ratio of
specific heats
Subsitute Eq. 7 into Eq. 6
Eq. 8:
gamma * p * dr / r = a^2 * dr
We can integrate the equation for the calorically perfect case
because the ratio of specific heats is a constant:
Eq. 8a:
gamma * p / r = a^2
The equation of state:
Eq. 9:
p = r * R * T
Eq. 9a:
p / r = R * T
where R is the gas constant, and T is the temperature.
Substitute Eq. 9a into Eq. 8a:
Eq. 10:
a^2 = gamma * R * T
Eq. 10a:
a = sqrt (gamma * R * T)
For the calorically imperfect case, we can not perform the simple integration
at Equation 8.
Mathematical models
based on a simple harmonic vibrator have been developed for the calorically
imperfect case..
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:
Eq. 11:
a = sqrt (R * T * {1 + (gamma  1) / ( 1 + (gamma1) * [(theta/T)^2 * e^(theta/T) /(e^(theta/T) 1)^2]) })
where theta is a thermal constant
equal to 5500 degrees Rankine, and T is the static temperature.
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