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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

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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