Speed of Sound: Definition, Formula, Affecting Factors

Sound waves flow across a medium by contracting and expanding the medium’s sections in alternating patterns. The speed of sound is defined as the pace at which sound waves propagate across a medium. Light and sound are waves, yet light is seen before its sound is heard! This is because the speed of sound is lesser than the speed of light.

Sound waves are longitudinal waves that need a material medium to travel. They cannot travel in a vacuum but can travel in solids, liquids and gases. A sound wave is a pressure disturbance that travels through a medium through particle-to-particle interaction. In this article, we will learn more about the speed of sound, sound waves with various examples. 

Speed of Sound Wave

The speed of sound is the distance through which sound waves travel in a given amount of time. Sound waves are longitudinal waves that need a material medium to travel. They cannot travel in a vacuum but solids, liquids, and gases. Speed of sound is used to describe the speed of sound waves in an elastic medium. The speed of sound waves in a different medium depends on the characteristics of the medium through which the propagation takes place. However, in a given medium, it remains constant.

Learn Effect Of Pressure And Humidity On Velocity Of Sound

Its speed does not depend on the wave’s characteristics or the force that generates it. Therefore, its propagation in a medium can be used to study its properties. The speed of sound is the distance travelled by a sound wave propagating through an elastic medium per unit of time. The speed of sound in a given medium depends on that medium’s density and elasticity properties. According to physics, the speed of sound is greater in the medium with greater elasticity and smaller density. Therefore, the speed of sound is maximum in solids and minimum in gases.

Newton’s Formula for the Speed of Sound

The velocity of a longitudinal wave through an elastic medium is:
\(v = \sqrt {\frac{B}{\rho }} \)
Where \(B\) is the Bulk modulus, and \(\rho \) is the density of the medium. Since a sound wave is a longitudinal wave, we can use this relation to derive the formula for the speed of sound through a medium.

Newton assumed that the propagation of sound through a medium is an isothermal process. He thought that the medium in which sound is propagating is in contact with surroundings, and any temperature changes will be neutralised by the surroundings.
Therefore, \(v = \sqrt {\frac{{{B_{{\rm{iso}}}}}}{\rho }} \;\)
For an isothermal process,
\(PV = {\rm{const}}\)
Differentiating both sides of the equation, we get:
\(PdV + VdP = 0\)
\(\therefore \,\,\,\,\frac{{dP}}{{dV}}V = – P\)
The volume elasticity of a medium is,
\({B_{{\rm{iso}}}} = – V\frac{{dP}}{{dV}}\)
Thus, from above, \({B_{{\rm{iso}}}} = P\)
Therefore, the speed of sound, According to Newton’s formula:
\(v = \sqrt {\frac{P}{\rho }} \)
The speed of sound in air,
At Normal-temperature and Pressure, the density of air is \(\rho = 1.293\,\;{\rm{kg}}/{{\rm{m}}^3}\)
Atmospheric Pressure \(P = 1.013 \times {10^5}\;{\rm{N}}{{\rm{m}}^{ – 2}}\)
Substituting the values,
\(v = \sqrt {\frac{{1.013 \times {{10}^5}{\rm{\;N\;}}{{\rm{m}}^{ – 2}}}}{{1.293{\rm{\;kg\;}}{{\rm{m}}^{ – 3}}}}} \)
\(\therefore \,\,\,v = 280\;{\rm{m\;}}{{\rm{s}}^{ – 1}}\)
But this value is almost \(16\%\) less than the actual value of the speed of sound in air, i.e., \(320\,\,{\rm{m}}{{\rm{s}}^{ – 1}}.\) This means that this formula required some corrections.

Laplace Correction for Newton’s Formula

He corrected Newton’s formula by assuming that no heat exchange occurs as the motion of compression and rarefaction takes place very fast. Thus, the temperature does not remain constant, and the sound wave propagation through the air is an adiabatic process.
Thus, for an adiabatic process,
\(P{V^\gamma } = {\rm{constant}}\)
Where \({\rm{\gamma }}\)  is the ratio of specific heat capacity and is equal to, \({\rm{\gamma }} = {C_P}/{C_V}\)
\({C_p}:\) specific heat at constant pressure
\({C_v}:\) specific heat at constant volume
Differentiating both the sides we get-
\(\gamma P{V^{\gamma – 1}}dV + {V^\gamma }dP = 0\)
Dividing both the sides by \({V^{\gamma – 1}}\)
\(dP + \gamma P{V^{ – 1}}dV = 0\)
\(P\gamma = – V\frac{{dP}}{{dV}} = B\)
The velocity of sound is given by \(\;v = \sqrt {\frac{B}{\rho }} \)
Substituting \({\rm{\;}}B = \gamma P\) in the above equation, we get:
\(v = \sqrt {\frac{{\gamma P}}{\rho }} \)

Speed of Sound in Air

Using Laplace correction to Newton’s formula, at Normal Temperature and Pressure,
The formula to calculate the speed of the sound is given by: \(v = \sqrt {\frac{{\gamma P}}{\rho }} \)
Where,
The ratio of specific heat capacity: \(\;\gamma = 1.4\)
Atmospheric pressure \(P{\rm{\;}} = {\rm{\;}}1.1013 \times {10^5}{\rm{\;N\;}}{{\rm{m}}^{ – 2}}\)
The density of air is \(\rho = 1.293{\rm{\;kg\;}}{{\rm{m}}^{ – 3}}\)
Substitute the values in the above formula, we get:
\(v = \sqrt {\frac{{1.4 \times 1.013 \times {{10}^5}{\rm{\;N\;}}{{\rm{m}}^{ – 2}}}}{{1.293{\rm{\;kg\;}}{{\rm{m}}^{ – 3}}}}} \)
\(\therefore \,\,v = 332\;{\rm{m\;}}{{\rm{s}}^{ – 1}}\)
This value of the speed of sound in the air was nearly the same as the speed of sound obtained experimentally. Hence Laplace’s correction was taken up in Newton’s formula to calculate the speed of sound.

Factors Affecting Speed of Sound

The speed of any wave depends upon the properties of the medium through which the wave is travelling.
(i) Effect of Pressure: As calculated above, the formula for the speed of sound in a gas.
\(v = \sqrt {\frac{{\gamma P}}{\rho }} = \sqrt {\frac{{\gamma RT}}{m}} \)
Where \(R\) is the gas constant, \(T\) is the temperature, and \(m\) is the mass.
As the pressure of the medium changes, its density also changes, such that the ratio of the two quantities remains unchanged. Therefore, \(\left( {\gamma P/\rho } \right)\) remains constant at a constant temperature.
Hence, there is no effect of pressure on the speed of the sound waves.
(ii) Effect of Temperature: The velocity of the sound wave as per Laplace’s correction to the formula was calculated as:
\(v = \sqrt {\frac{{\gamma P}}{\rho }} = \sqrt {\frac{{\gamma RT}}{m}} \)
\(\therefore \,\,v \propto \sqrt T \)
The speed of sound in a gas is directly proportional to the square root of its absolute temperature.
\(\frac{{{v_1}}}{{{v_2}}} = \sqrt {\frac{{{T_1}}}{{{T_2}}}} \)
Therefore, the speed of sound waves through a medium increases with the rise in temperature of that medium.

If \({v_0}\) and \({v_t}\) be the velocities of sound in air at \(0{\rm{\;}}^\circ {\rm{C}}\) and \(t{\rm{\;}}^\circ {\rm{C}},\) then:
\({v_t} = {v_0}{\left( {1 + \frac{t}{{273}}} \right)^{1/2}}\)
\({v_t} = {v_0} + 0.61{\mathop{\rm t}\nolimits} \)
Hence, the velocity of sound in air increases by \(0.61{\rm{\;m}}/{\rm{s\;}}\) for every \(1{\rm{\;}}^\circ {\rm{C\;}}\)  rise in temperature.
(iii) Effect of Density: According to the formula of sound in a gaseous medium:
\(v \propto \frac{1}{{\sqrt \rho }}\)
The velocity of sound in a gas is inversely proportional to the square root of the density of the gas.
\(\frac{{{v_1}}}{{{v_2}}} = \sqrt {\frac{{{\rho _1}}}{{{\rho _2}}}} \)
This means that as the density of the medium increases, the speed of sound through it decreases.
(iv) Effect of Humidity: The speed of sound in moist air is greater than the speed of sound in dry air for the same pressure. We know that the density of moist air is less than that of dry air; an increase in humidity means an increase in moist air. This implies that as humidity increases, the speed of sound in the air increases.

Speed of Sound in Different Media

Sound waves are mechanical waves, and mechanical waves can only travel through matter. Therefore, the matter through which the waves travel is called the medium. Solids, liquids and gases differ in the arrangements of their particles and the forces between the particles. Particles of matter are closest together in solids and farthest apart in gases. When particles are closer together, they can more quickly pass the energy of vibrations to nearby particles. 

1. Speed of sound in Solids: Sound waves are characterised by the motion of particles in the medium and are called mechanical waves. The molecules density is maximum in solids and least in gases. This means that solid’s particles are placed closer to each other than the particles in liquids or gases. Due to this, an individual particle in a solid can collide with its neighbouring particle in lesser time. That’s why a disturbance can travel much more easily and quickly through solids, and hence the speed of sound is maximum in solids compared to liquids or gases. For example, the speed of sound through copper and aluminium is \(6420{\rm{ }}\,{\rm{m/s}}.\) The sound waves travel over \(17\) times faster through steel than through air.

2. Speed of sound in Liquids: The particles in liquids are less tightly packed than the particles in solids. Thus the density of molecules is lesser in liquids than solids or gases. Thus, the distance between molecules of liquids is greater than the distance between molecules of solids but still lesser than the distance between molecules of gases. This is why the speed of sound in liquids is less than the speed of sound in solids but is greater than the speed of sound in gases.
The speed of sound in water is about \(1450{\rm{ }}\,{\rm{m/s}}.\) although it can vary with the temperature and density of the water. It is almost four times the speed of sound in the air.

3. Speed of sound in Gases: Molecules of gases are farthest apart from solids or liquids molecules. The density of gases is considered to be uniform. That is why the speed of sound in gases is independent of the medium.

4. Speed of sound in a vacuum: A vacuum is a space devoid of matter. A sound wave is a longitudinal wave that needs a medium for its propagation. Thus, sound waves do not travel through a vacuum, or the sound waves speed is zero.

Mach Number

The Mach Number is the ratio of flow velocity after a certain limit of the sound’s speed. In simple words, it is the ratio of the speed of a body to the speed of sound in the surrounding medium. For simplification, we take the speed of sound that can be equated to Mach Number \(1\) speed. Thus, Mach \(0.75\) will be \(75\%\) of the speed of sound. Also, Mach Number \(1.65\) will be \(65\%\) faster than the speed of sound. The Mach Number can be calculated using:
\(M = v/c\)
Where, \(v\) is the given speed, and \(c\) is the speed of sound in the given medium.
Subsonic: A vehicle traveling slower than the speed of sound \(\left( {M < 1} \right)\) is said to be flying at subsonic speeds.
Supersonic: A vehicle traveling faster than the speed of sound \(\left( {M > 1} \right)\) is said to be flying at supersonic speeds.
Sound barrier: The term sound barrier is often associated with supersonic flight. In particular, “breaking the sound barrier” is the process of accelerating through Mach \(1\) and going from subsonic to supersonic speeds.

Solved Problems

Question: Calculate the temperature at which the speed of sound in the air becomes double its value at \(127{\,^{\rm{o}}}{\rm{C}}.\)
Solution: Let the initial speed of sound in the air be \(v.\)
The final speed, \(v’ = 2v\)
The initial temperature of the air, \(T = 127\;{\,^{\rm{o}}}{\rm{C}} = 400\;{\rm{K}}\)
Let \(T’\) be the final temperature.
We know, for speed of sound in air, \(v \propto \sqrt T \)
Thus, \(\frac{{v’}}{v} = \sqrt {\frac{{T’}}{T}} \)
\(\frac{{2v}}{v} = \sqrt {\frac{{T’}}{{400\;{\rm{K}}}}} \)
\(T’ = 1600\;{\rm{K}} = 1327{\,^{\rm{o}}}{\rm{C}}\)

Summary

The speed of sound is the distance through which sound waves travel in a given amount of time. Sound waves are longitudinal waves that need a material medium to travel. These cannot travel in a vacuum, but solids, liquids, and gases. The speed of sound waves is maximum in solids. In liquids, the speed of sound is less as compared to that in solids but it is greater than the speed of sound through gases.

Newton calculated the formula speed of sound, assuming the transfer of sound waves through a medium to be an isothermal process. According to this formula, \(v = \sqrt {\frac{P}{\rho }}.\) The results obtained by using this formula did not match with the actual value of the speed of sound. Laplace corrected this formula, assuming the transfer of sound waves to be an adiabatic process and gave the formula, \(v = \sqrt {\frac{{\gamma P}}{\rho }} ,\) where \({\rm{\gamma }}\) is the ratio of specific heat capacity.
1. The speed of sound waves through a medium increases with the rise in temperature of that medium.
2. As humidity increases, the speed of sound in the air increases.
3. As the density of the medium increases, the speed of sound through it decreases.
4. There is no effect of pressure on the speed of the sound waves.

FAQs

Q.1. How fast does sound travel in the air in mph?
Ans: The speed of sound in air varies with the temperature. For example, the speed of sound at sea level, assuming the atmospheric temperature of \(59{\,^{\rm{o}}}{\rm{F}}\left( {15{\,^{\rm{o}}}{\rm{C}}} \right),\) is \(761.2{\rm{ }}\,{\rm{mph }}\left( {1,225\,{\rm{ km/h}}} \right).\)

Q.2. What is the exact speed of sound?
Ans: The speed of sound depends on the medium in which it is propagating. Two properties of the medium on which speed depends are elasticity and density. At \(20{\rm{ ^\circ C}},\) the speed of sound in air is about \(343\) metres per second or \(1,235\,{\rm{ km/h}}.\)

Q.3. What is the fastest Mach speed?
Ans: Mach Number was named after the Austrian physicist Ernst Mach. Mach \(1\) is the speed of sound, which is approximately \(760\) miles per hour at sea level. NASA’s \(X – 43A\) scramjet set a world speed record for a jet-powered aircraft by travelling at Mach \(9.6,\) or nearly \(7,000{\rm{ }}\,{\rm{mph}}.\) Thus, the fastest Mach speed recorded to date is Mach \(9.6.\)

Q.4. What is the fastest human-crewed jet in the world?
Ans: The aircraft  North American \(X – 15\) has the current world record for the fastest human-crewed aircraft. Its maximum speed was Mach \(6.70\) (about \(7,200\,{\rm{ km/h}}\)), which it attained on the \(3\rm{rd}\) October, \(1967\) thanks to its pilot William J.

Q.5. Is Mach the speed of sound?
Ans: Mach Speed is when an object moves faster than the speed of sound. AT NTP of \({\rm{68}}{\,^{\rm{o}}}{\rm{F}},\) this is \(768{\rm{ }}\,{\rm{mph}}\) or \(343{\rm{ }}\,{\rm{m/s}}\) or \(1,235{\rm{ }}\,{\rm{km/h}}{\rm{.}}\) Mach \(1\) refers to the point at which an aircraft exceeds the speed of sound, creating a sonic boom. Mach \(2\) refers to a point when aircraft travels at twice the speed of sound and so on.

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