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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. 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 + (gamma-1) * [(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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Editor: Tom Benson
NASA Official: Tom Benson
Last Updated: May 07 2021

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