
Speed of Sound Derivation


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.
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
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)
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