2.5Formulas for the Speed of Sound

The speed of the sound can be formulated using pressure, density, temperature, or bulk modulus. The following sections show three formulas for calculating the speed of sound.

2.5.1Formula Using Pressure

In the previous section, the speed of the sound c was formulated using pressure and density as:

r

c ¼ γPo

ρo

where γ is the ratio of specific heats and the approximate value of γ for air is:

γ ¼ 1.4 (for air at any temperature)

Example 2.1 Air at 1 atm, 15 C has a density:

hKgi

ρo ¼ 1:225 m3

Calculate the speed of sound in air at 1 atm, 15 C based on the pressure, density, and ratio of specific heats. (Hint: Air has the ratio of specific heats γ ¼ 1.4.)

Example 2.1 Solution The pressure at 1 atm is:

Po ¼ 1:01 ∙ 105 ½Pa& ð1 atmÞ

The speed of sound c at 15 C for air can be calculated based on the pressure Po, density ρo, and ratio of specific heats γ as:

where:

hKgi

ρo ¼ 1:225 m3

γ ¼ 1:4

Note that in the formula for the speed of sound above, the speed c is a function of pressure Po and density ρo. However, changing the pressure does not change the speed of sound because the effect from density ρo cancels out the effect from pressure Po.

For example, at higher altitudes, the decrease of density is linearly proportional to the decrease of air pressure, and the speed of sound here will be the same as at sea level.

As a result, the temperature is the most significant factor in the speed of the sound and will be shown in the next section.

2.5.2Formula Using Bulk Modulus

The speed of sound is related to the stiffness of the media. The coefficient of stiffness is also called bulk modulus and is defined as the ratio of pressure increase to volume decrease as:

!dP ¼ γ dVa Po Va

Substituting the equation above in the definition of bulk modulus yields:

B ¼ γPo

The formula for the speed of sound (based on bulk modulus) can be obtained by substituting the bulk modulus into the formula for the speed of sound (based on pressure) as:

r

c ¼ γPo

ρo r

! c ¼

B

ρo

where B is the bulk modulus and ρo is the average density of the media.

The formula above for the speed of sound using bulk modulus is valid for both gases and liquids.

Example 2.2 Air at 1 atm, 15 C has a bulk modulus B and density ρo as:

B ¼ 1:414 105 ½Pa&

hKgi

ρo ¼ 1:2 m3

Calculate the speed of sound in air at 1 atm, 15 C based on the given bulk modulus B and density ρo.

Example 2.2 Solution The speed of sound c at 15 C for air can be calculated based on the given bulk modulus B and density ρo as:

where

B ¼ 1:414 105 ½Pa&

hKgi

ρo ¼ 1:2 m3

The speed of sound in both air and water can be calculated using the same formula in this section. The speeds of sound in air and water are tabulated below:

2.5.3Formula Using Temperature

The ideal gas law states that:

PV ¼ nNAkBT ¼ nRT

where

n: number of moles

NA: number of particles per mole (Avogadro constant) kB: Boltzmann’s constant

R: ideal gas constant (R ¼ NAkB)

T: absolute temperature (K¼273.15+C)

The formula for the speed of sound (based on temperature) can be obtained by substituting the ideal gas law into the formula for the speed of sound (based on pressure) as:

where

γ: ratio of specific heats ¼1.4 (for air, N2, and O2 at any temperature) M: molar mass of the gas (mass per moles)

T: absolute temperature (K¼273.15+C)

Example 2.3 The average molar mass for air at 15 C is 0.02895 molKg . Calculate the speed of sound at 15 C based on the ratio of specific heats γ , molar mass M, ideal gas

constant R, and temperature T.

Example 2.3 Solution The speed of sound in air at 15 C can be calculated based on the ratio of specific heats , molar mass M, ideal gas constant R, and temperature T as:

where: