__What is the speed of Sound?__

Scientist Newton first stated a theoretical expression for *speed* of sound in a gaseous medium. Newton’s theory for speed of sound in gases is based on the following assumptions.

- Longitudinal sound waves travel through a gaseous medium under
*isothermal conditions*. - During propagation of sound,
*longitudinal waves*produce compression and rarefaction in the medium. - A small heat is produced in the
*compression region*and cooling in the*rarefaction region*of medium. - This small amount of heat which produced in the compression region is rapidly conducted to the rarefaction region which is comparatively cooler.
- Thus,
*isothermal conditions*are maintained throughout the propagation process in gaseous medium.

Thus, based on above assumptions, the speed of sound in a gas will be –

v = \sqrt { \frac { k _ { iso } }{ \rho } }

Here, ( k _ {iso} ) is the *bulk modulus of elasticity* for air or gas in isothermal conditions.

For an isothermal change in a gas, we have got the relation \quad P V = \text {Constant}

Differentiating this expression, we will get \quad P dV + V dP = 0

Or, \quad P = - \left ( \frac { d P }{ d V / V } \right )

= \left ( \frac { \text {Change in pressure}}{\text {Volumetric strain}} \right )

But, \quad \left ( \frac { \text {Change in pressure}}{\text {Volumetric strain}} \right ) is called *bulk modulus of rigidity.* So, \quad P = k _ { iso }

Hence, speed of sound in a gaseous medium will be –

v = \sqrt { \frac {k _ {iso}}{ \rho}} = \sqrt {\frac { P }{ \rho } }

Or, \quad v = \sqrt { \frac { P }{ \rho } }

*This is called ***Newton’s formula** for speed of sound in a gaseous medium.

__Speed of Sound in Air__

From Newton’s formula, speed of sound in a gaseous medium is given by –

v = \sqrt { \frac { P }{ \rho } }

For air, this expression can be written as –

v = \sqrt { \frac { \text {Atmospheric pressure}}{ \text {Density of air}} }

At STP , atmospheric pressure \quad P = 1.013 \times 10 ^ 5 \ \text {N-m} ^ { - 2 } , Density of air \quad \rho = 1.293 \ \text {kg-m} ^ { - 3 }

Putting these values in the above expression –

v = 280 \ \text {m-s} ^ { - 1}

*Hence, according to Newton’s formula the velocity of sound in air is ( 280 \ \text {m-s} ^ { - 1}) *

__Laplace’s Correction__

Based on Newton’s assumptions, the velocity of sound in air is ( 280 \ \text {m-s} ^ { - 1} ) . But this value is less than the experimental value of ( 331 \ \text {m-s} ^ { - 1 } ) . Hence, Newton’s formula needs some corrections.

Another scientist Laplace suggested that –

- When sound travels through a gas or air, the average temperature in the compression regions rises and in the rarefaction regions falls.
- Also a gas or air is a poor conductor of heat.
- The compression and rarefaction in the medium occur so rapidly that heat generated in the compression regions does not get sufficient time to conduct into the rarefaction region.
- As such
*isothermal condition*of medium is not maintained as suggested by Newton’s formula, instead*adiabatic conditions*are prevailed in the medium. - Hence, Newton’s formula for speed of sound need correction.

As suggested by Laplace’s corrections, the speed of sound in a gas or air will be –

v = \sqrt { \frac { k _ { adia } }{ \rho } }

Here ( k _ {adia} ) is the *bulk modulus of elasticity* for air or gas in adiabatic conditions.

For an adiabatic change in a gas, we have the relation –

P V ^ { \gamma } = \text {Constant}

Differentiating both sides, we will get –

P ( \gamma V ^ { \gamma - 1 } ) d V + V ^ { \gamma } dP = 0

Dividing the equation by ( V^{\gamma - 1} ) , we will get –

\gamma P dV + V dP = 0

Or, \quad \gamma P = - \left ( \frac { d P }{ d V / V } \right ) = k _ { adia }

Here, \left [ \gamma = \frac { C _ { p } }{ C _ { v } } \right ] is the ratio of two *specific heats* of a gas.

Hence, the Laplace’s formula for speed of sound in a gas will be –

v = \sqrt { \frac { \gamma P }{ \rho } }

Or, \quad v = \sqrt { \frac { \gamma \times \text {Atmospheric pressure}}{\text {Density of air}} }

*This modification in Newton’s formula is popularly known as Laplace’s correction.*

For air, the value of \left ( \gamma = \frac { 7 }{ 5 } \right ) .

Therefore, the speed of sound in air will be –

v = \sqrt { \frac { 7 }{ 5 } \times 280} = 331.3 \text {m-s} ^ { - 1}

This value is close to the experimental value of ( 331 \text {m-s} ^ { - 1 } )

*Hence, Laplace’s correction in Newton’s formula is justified and acceptable.*

__Factor’s affecting speed of Sound__

Speed of sound in a given gaseous medium depends upon different factors. These factors are –

- Pressure of gas.
- Density of gas.
- Humidity or moisture content in gas.
- Temperature of gas.
- Wind speed etc.

__Effect of Pressure__

The speed of sound in a gaseous medium is given by Laplace’s formula –

v = \sqrt {\frac {\gamma P}{\rho}}

For a gas at constant temperature by *Boyle’s law* –

PV = \text {Constant}

Also mass of a gas \left ( m = \rho V \right ) or \left ( V = \frac {m}{\rho} \right )

Therefore, \quad PV = \frac {P \ m}{\rho} = \text {Constant}

Since ( m ) is a constant, so \quad \left ( \frac {P}{\rho} \right ) = \text {Constant}

Therefore, when pressure of a gas changes, density also changes in the same ratio so that the ratio \left ( \frac {P}{\rho} \right ) remains unchanged.

*Hence, pressure change has no effect on the speed of sound in a gas.*

__Effect of Density__

Suppose two gases have the same pressure ( P ) and have same value of ( \gamma ) .

If ( \rho_1 ) and ( \rho_2 ) are their densities, then the speed of sound through them will be –

- v_1 = \sqrt {\frac {\gamma P}{\rho_1}}
- v_2 = \sqrt {\frac {\gamma P}{\rho_2}}

Therefore, \quad \left ( \frac {v_1}{v_2} \right ) = \sqrt {\frac {\rho_2}{\rho_1}}

*Hence, at constant pressure the speed of sound in a gas is inversely proportional to the square root of its density.*

__Effect of Humidity__

The speed of sound in air is given by Laplace’s formula as –

v = \sqrt {\frac {\gamma P}{\rho}}

Or, \quad v \propto \left ( \frac {1}{\sqrt \rho} \right ) .

- At STP , density of water vapour is \left ( 0.8 \ kg \ m^{-3} \right )
- At STP , density of dry air is \left ( 1.293 \ kg \ m^{-3} \right )

Hence, density of water vapour is less than the density of dry air. So, the presence of moisture content in air results in decrease of density.

*So sound wave travels faster in moist air than in dry air.*

__Effect of Temperature__

From *ideal gas equation*, for one mole of a gas we have the relation –

PV = RT

Let, ( M ) * *is the molecular mass and ( \rho ) is the density of gas. Then, \quad \rho = \left ( \frac {M}{V} \right )

Or, \quad V = \left ( \frac {M}{\rho} \right )

Putting this relation in gas equation, we get –

P \left ( \frac {M}{\rho} \right ) = RT

Or, \quad \left ( \frac {P}{\rho} \right ) = \left ( \frac {RT}{M} \right )

Therefore, speed of sound in gas will be \quad v = \sqrt {\frac {\gamma RT}{M}}

Or, \quad v \propto \sqrt T

*Therefore, speed of sound in a gas is directly proportional to the square root of absolute temperature.*

### Temperature coefficient of Sound speed

*Temperature coefficient for sound speed in air is defined as the increase in velocity of sound for ( 1 \degree C ) or ( 1 \degree K ) rise in temperature of air.*

Let,

- ( v_t ) is the speed of sound in air at temperature ( t \degree C ) .
- ( v_0 ) is the speed of sound in air at temperature ( 0 \degree C ) .

Then –

\left ( \frac {v_t}{v_0} \right ) = \sqrt {\frac { \left ( t + 273 \right ) }{ \left ( 0 + 273\right ) }}

Or, \quad v_t = v_0 \sqrt {\frac { \left ( t + 273 \right ) }{273}} = v_0 \left [1 + \frac {t}{273} \right ]^{\frac{1}{2}}

= v_0 \left [1 + \frac {t}{546} \right ]

So, \quad \left ( v_t - v_0 \right ) = v_0 \left ( \frac {t}{546} \right )

But speed of sound at ( 0 \degree C ) is ( 332 \text {m-s}^{-1} )

Therefore, \quad \left ( v_t - v_0 \right ) = 332 \left [ \frac {t}{546} \right ] = 0.61 \times t

*The quantity ( 0.61 ) is called the temperature coefficient of sound.*

When, ( t = 1 \degree C ) , Then \quad \left ( v_t - v_0 \right ) = 0.61 \text {m-s}^{-1}

*Therefore, the coefficient for the speed of sound in air is ( 0.61 \text {m-s}^{-1} ) *

__Effect of Wind speed__

Velocity of sound in air is affected by wind velocity.

Suppose that, wind travels with speed ( \omega ) at an angle ( \theta ) with the direction of propagation of sound. Then component of wind velocity in the direction of sound wave is \left ( \omega \cos \theta \right )

Therefore, resultant velocity of sound will be –

v = \left ( v + \omega \cos \theta \right )

Similarly, when wind blows in direction of opposite to sound, then resultant velocity will be –

v = \left ( v - \omega \cos \theta \right )

See numerical problems based on this article.