Speed of Sound

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.