Speed of Sound Waves in Air - Formula and Equation of Speed for JEE

A sound wave is a pressure disturbance that propagates through a medium via particle-to-particle interaction. When one particle is disturbed, it exerts a force on the next adjacent particle, causing that particle to be disturbed and the energy to be transported through the medium. As sound waves travel through a medium, they alternately contract and expand the parts of the medium through which they travel.

The speed of sound in air, like the speed of any wave, refers to how quickly the disturbance is passed from particle to particle. While frequency refers to the number of vibrations produced by a single particle per unit of time, speed refers to the distance travelled by the disturbance per unit of time.

What is Sound? 

Sound is defined in Physics as a vibration that travels as an acoustic wave through a transmission medium such as a gas, liquid, or solid. It is the reception of such waves and their perception by the brain in human physiology and psychology. Only acoustic waves with frequencies in the 20 Hz to 20 kHz range, known as the audio frequency range, elicit auditory perception in humans. These represent sound waves with wavelengths ranging from 17 metres (56 feet) to 1.7 centimetres in air at atmospheric pressure (0.67 in). Ultrasound is defined as sound waves above 20 kHz that are inaudible to humans. Infrasonic sound refers to sound waves with frequencies less than 20 hertz.

Characteristics of Sound

Sound waves have five main characteristics: wavelength, amplitude, frequency, time period, and velocity.

• Wavelength

The wavelength is possibly the most important characteristic of sound waves. Sound is a longitudinal wave that experiences compressions and rarefactions as it travels through a medium. The wavelength is the distance that one wave travels before repeating itself. It is the distance between the centres of two consecutive rarefactions or compressions, or the combined length of a compression and the adjacent rarefaction.

• Amplitude

The amplitude of a given wave is its size. Consider the height of the wave as opposed to its length. The amplitude is more precisely defined as the maximum displacement of the particles disturbed by a sound wave as it travels through a medium.

• Frequency

The frequency of a sound is the number of sound waves produced per second. A low-frequency sound contains fewer waves than a high-frequency sound. Sound frequency is measured in hertz (Hz) and is independent of the medium through which the sound travels.

• Time Period

The time period is nearly inverse to the frequency. It is the amount of time required to generate a single complete wave, or cycle. Each vibration of the vibrating body that produces the sound corresponds to a wave.

• Velocity

The velocity of the wave, also known as the speed, is the amount of distance travelled by a wave in one second in metres per second.

What is the Speed of Sound?

As sound waves travel through a medium, they contract and expand the medium's parts. The distance travelled by any sound per unit of time is defined as the speed of sound. In the following section, we will learn how to calculate the speed of sound in various mediums. Sound wave propagation in a dynamic environment is known as the speed of sound. 

The speed of sound is determined by the properties of the medium in which the transmission occurs. The term “speed of sound” refers to the velocity of sound waves in an elastic medium. Hence, the speed of sound defines how quickly it can propagate in some medium.The speed of sound in a given medium is determined by its elasticity and density. The greater the elasticity and the lower the mass, the greater the sound speed. As a result, the speed of sound is greatest in solids and lowest in gases.

It is defined as the dynamic propagation of waves through various mediums. The speed of sound varies with the medium through which it travels. When we talk about sound speed, we mean the speed at which sound waves travel in an elastic medium.

The formula for the speed of sound according to Newton with respect to gases is as given:

Here P is the pressure of the gas and ρ is the density of the medium. 

Newton researched the propagation of sound waves through air. He assumed that the propagation process is isothermal. Since heat absorption and release are balanced during compression and rarefaction, the temperature remains constant throughout the process.

Boyle's law states that

PV = Constant

Where,

P is pressure

V is the volume of gas.

On differentiating the above equation, we get

PdV+VdP=0

 PdV = -VdP

 P = B

Where, 

$B=\dfrac{d P}{-\left(\dfrac{d V}{V}\right)}$

B is the bulk modulus of air.

The velocity of the sound wave can be written as

$v=\sqrt{\dfrac{B}{\rho}}$

Thus substituting B =P we get

$v=\sqrt{\dfrac{P}{\rho}}$

𝜈 represents the speed of sound

P represents the pressure of the gas.

$\rho$ reflects the density of the medium through which sound.

Hence, this is the speed equation and for the velocity of sound in air, we will have a particular direction.

Laplace Correction for Newton’s Formula

Laplace modified Newton's formula by assuming that no heat exchange occurs because compression and rarefaction happen so quickly. As a result, the temperature does not remain constant, and sound wave propagation in air is an adiabatic process.

For adiabatic processes,

$P V^\gamma=\text { constant }$

Where,

$\gamma=\dfrac{C_{p}}{C_{v}}$

$\gamma$ is used as the adiabatic expansion coefficient.

Cp specific heat for constant pressure

Cv specific heat for a constant volume.

Differentiating both the sides we get

$V^\gamma d P+P\gamma V V^{-1} d V=0$

Dividing both the sides by $V^{\gamma-1}$

$\begin{align} &V d P+P \gamma V d V=0 \\ &P \gamma=-\dfrac{d P}{\left(\dfrac{d V}{V}\right)}=B \end{align}$

The velocity of sound is given by

$v=\sqrt{\dfrac{B}{\rho}}$

Substituting $B= \gamma P$ in above equation, we get:

$v=\sqrt{\dfrac{\gamma P}{\rho}}$

Factors Affecting the Speed of Sound

The following factors have a significant impact on the speed of the sound:

• Effect of Pressure

When the pressure varies at a constant temperature, the density varies correspondingly, causing the ratio  ($\dfrac{P}{\rho}$) to become constant, as $v=\sqrt{\dfrac{\gamma P}{\rho}}$

For a constant temperature, this means that the speed of sound is independent of pressure. 

• Effect of Temperature

As $v=\sqrt{\dfrac{\gamma P}{\rho}}$

The speed of sound varies directly to the square root of temperature in Kelvin.

Let v0 be the speed of sound at the temperature at 0° C or 273 K and v be the speed of sound at any other temperature T (in kelvin), then

$\begin{align}\dfrac{v}{v_{0}}&=\sqrt{\dfrac{T}{273}}\\ \dfrac{v}{v_{0}} &=\sqrt{\dfrac{273+t}{273}} \\ v &=v_{0} \sqrt{1+\dfrac{t}{273}} \end{align}$

Using binomial expansion,

$\cong v_{0}\left(1+\dfrac{t}{546}\right)$

Note that when the temperature is increased, the molecules will vibrate faster due to gain in thermal energy and hence, speed of sound increases.

• Effect of Density

Consider two gases with different densities that are both at the same temperature and pressure. Then the sound speeds in the two gases are calculated.

$\begin{align}&v_{1}=\sqrt{\dfrac{\gamma_{1} P}{\rho_{1}}} \\ &v_{2}=\sqrt{\dfrac{\gamma_{2} P}{\rho_{2}}} \end{align}$

Taking the ratio of both, we get

$\dfrac{v_{1}}{v_{2}}=\sqrt{\dfrac{\gamma_{1} \rho_{2}}{\gamma_{2} \rho_{1}}}$

For gases having same $\gamma$, we get

$\dfrac{v_{1}}{v_{2}}=\sqrt{\dfrac{\rho_{2}}{\rho_{1}}}$

As a result, the velocity of sound in a gas is inversely proportional to the square root of the gas's density

• Effect of Moisture (Humidity)

We know that the density of moist air is 0.625 that of dry air, implying that the presence of moisture in air (increase in humidity) reduces its density. As a result, as humidity rises, the speed of sound also increases. As we have $v=\sqrt{\dfrac{\gamma P}{\rho}}$

Let $\rho_1,~v_1$ and $\rho_2,~v_2$ be the density and speeds of sound in dry air and moist air, respectively. Then 

$\dfrac{v_{1}}{v_{2}}=\dfrac{\sqrt{\dfrac{\gamma_{1} P}{\rho_{1}}}}{\sqrt{\dfrac{\gamma_{2} P}{\rho_{2}}}}$

if $\gamma_{1}=\gamma_{2}$

$\dfrac{v_{1}}{v_{2}}=\sqrt{\dfrac{\rho_{2}}{\rho_{1}}}$    

Since P is the total atmospheric pressure, so 

$\dfrac{\rho_{2}}{\rho_{1}}=\dfrac{P}{p_{1}+0.625 p_{2}}$

Where p1 and p2 are the dry air and water vapour partial pressures, respectively. Then $v_{1}=v_{2} \sqrt{\dfrac{P}{p_{1}+0.625 p_{2}}}$

• Effect of Wind

Wind also has an effect on the speed of sound. The speed of sound increases in the direction of the wind blowing, while it decreases in the opposite direction.

Sound Speed in Different Media

Sound can travel through various media, and the following is how it does so:

• Sound’s Speed in Solid

The collision of different molecules and particles causes sound to travel through solids. Solids have a higher density than other mediums, resulting in a high sound speed. The speed of sound in solids is approximately 6000 m/s.

• Sound’s Speed in Liquid

Liquids have a lower density than solids but a higher density than gases. As a result, the speed of sound in liquids falls somewhere between the speeds of solids and gases.

• Sound’s Speed in Gases

The speed of sound in gases is independent of the medium. This is due to the uniformity of gas density regardless of its type.

• Sound’s Speed in a Vacuum

Sound doesn't travel through a vacuum, so its speed is not zero. This is due to the absence of particles in a vacuum. There is no sound wave propagation in a vacuum.

Conclusion

In short, sound is nothing but a form of energy. The distance travelled per unit time by a sound wave travelling in an elastic medium is known as the speed of sound. The speed of sound in a given medium depends on the elasticity and density of that medium. We have discussed the introduction, sound definition, characteristics of sound, speed of sound, factors that affect the sound, and more. This article explains all the details related to sound and the speed of sound along with vital formulae.