Speed of Sound Formulas

Calculation formulas for velocity of sound in gas, fluid or solid

As a consequence of the compressibility of gases, fluids or solids, a disturbance introduced in some point of a substance will propagate through the substance with a finite velocity.

Acoustic Velocity and Speed of Sound

The velocity at which these small disturbances will propagate through the medium is called the Acoustic Velocity or the Speed of Sound.

The acoustic velocity is related to the change in pressure and density of the substance and can be expressed as

c = (dp / dρ)^1/2 (1)

where

c = sound velocity (m/s, ft/s)

dp = change in pressure (Pa, psi)

dρ = change in density (kg/m³, lb/ft³)

Speed of Sound in Gases, Fluids and Solids

The acoustic velocity can alternatively be expressed as

c = (E / ρ)^1/2 (2)

where

E = bulk modulus elasticity (Pa, psi)

This equation is valid for liquids, solids and gases. The sound travels faster through media with higher elasticity and/or lower density. If a medium is not compressible at all - incompressible - the speed of sound is infinite (c ≈ ∞).

Speed of Sound in Gases

Since the acoustic disturbance introduced in a point is very small, the heat transfer can be neglected and for gases assumed isentropic. For an isentropic process the ideal gas law can be used and the speed of sound can be expressed as

c = (k p / ρ)^1/2

= (k R T)^1/2 (3)

where

k = ratio of specific heats (adiabatic index)

p = pressure (Pa, psi)

R = gas constant

T = absolute temperature (^oK, ^oR)

For an ideal gas the speed of sound is proportional to the square root of the absolute temperature.

Example - Speed of Sound in Air

The speed of sound in air at 0 ^oC and absolute pressure 1 bar can be calculated as

c = (1.4 287 273)^1/2 = 331.2 (m/s)

where

k = 1.4

and

R = 287 (J/K kg)

The speed of sound in air at 20 ^oC and absolute pressure 1 bar can be calculated as

c = (1.4 287 293)^1/2

= 343.1 (m/s)

Example - Speed of Sound in Water

The speed of sound in water at 0 ^oC can be calculated as

c = (2.06 10⁹/999.8)^1/2

= 1435.4 (m/s)

where

E_v= 2.06 10⁹ (N/m²)

and

ρ = 999.8 (kg/m³)

• Speed of Sound in Water - Speed of sound in water at different temperatures - imperial and SI units.

Speed of Sound in Solids

• Speed of Sound in some common Solids
• Elastic Properties and Young Modulus for some Materials
• The Bulk Modulus for Elasticity
• Material Properties

Subsonic and Supersonic speed

• If the Mach Number is below 1, the flow velocity is lower than the speed of sound - and the speed is subsonic.
• If the Mach Number is 1, the flow velocity is lower than the speed of sound - and the speed is transonic.
• If the Mach Number is above 1, the flow velocity is higher than the speed of sound - and the speed is supersonic.

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