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#### Longitudinal Nature of Sound waves:

- Sound waves are the longitudinal waves. It can be explained as follows.

- Consider vibrating tuning fork. When the prongs of the tuning fork at rest the layers of medium (air) are at normal pressure.
- Now let us assume that the prongs of the fork are vibrating and the prongs are parting away. During this process, the prong will push the layers of air one over other and the air in this region gets compressed. Thus in this region pressure is maximum. This region is called compression.
- When the prongs return back to their normal position the pressure also becomes normal. Now, when the prongs try to come near to each other the pressure on the layers of air decreases. And layers also get parted i.e. spacing between them increases. And this condition is called rarefaction.
- Again the prongs come back to their normal position and the pressure of medium again becomes normal. This process repeats continuously.
- Thus alternate compression and rarefaction are formed at the same point. Thus the particles of the medium vibrate about their mean position. There is no actual migration of medium particles. Thus sound waves move forward in the form of alternate compressions and rarefactions.

#### Sound Waves as Audible Longitudinal Waves:

- Longitudinal waves can have any possible frequencies but, a normal human ear can hear the frequencies in the range 20 Hz to 20kHz. Therefore the longitudinal waves having frequencies in the range 20 Hz to 20kHz are called audible longitudinal waves. These frequencies are also referred as the sonic frequencies.
- The frequencies 20 Hz and 20kHz are known as the limits of audibility for a normal human hearing.
- Frequencies less than 20 Hz are called infrasonic frequencies. While the frequencies more than 20kHz are called the ultrasonic frequencies.

**Concept of ****Isothermal Change and Adiabatic Change:**

- A change in volume of a gas carried out at a constant temperature of a gas is called isothermal change.
- A change in volume of a gas carried with a change in the temperature of a gas is called adiabatic change.

**Newton’s Formula for Velocity of Sound waves in Air:**

- Newton studied the propagation of longitudinal waves through isotropic media and obtained formula for speed of sound in the medium

Where, E = Coefficient of elasticity of medium

ρ = Density of the medium.

- In case of solid the elasticity, constant E is replaced by Y, The Young’s modulus of elasticity. In case of fluids i.e. liquids and gases the elasticity constant E is replaced by K, the Bulk modulus of elasticity. Hence the formula is changed for gases as

- Newton assumed the propagation of sound in air as an isothermal process. At compression, the temperature of the medium increases while the temperature at rarefaction decreases. But Newton said that there is enough time to exchange heat between the compression and rarefaction. Thus as a whole temperature of the medium remains constant and the propagation of the sound wave is an isothermal process.

For an isothermal process, the bulk constant (K) is equal to the pressure of the medium (P).

Therefore, the velocity of sound wave in the air is given by

At N.T.P. the density of air is 1.29 kg / m³. And P = 0.76 x 13600 x 9.8 N/m².

Therefore the velocity of sound in air is given by

v = 280 m/s

- But experimentally the velocity of sound is found to be 332 m/s. Thus there is a considerable difference between the actual velocity of sound in air and calculated velocity of sound in air. Hence Newton’s formula for velocity requires correction.

**Laplace’s Correction to Newton’s Formula for Velocity of Sound Waves in Air:**

- Newton assumed the propagation of sound in air as an isothermal process. At compression, the temperature of the medium increases while the temperature at rarefaction decreases.
- Laplace corrected the formula assuming the process of propagation of sound as an adiabatic process. He said that formation of compression and rarefaction is so rapid that there is not enough time to exchange heat between compression and rarefaction. In the adiabatic process, the heat of system remains the same. Thus no heat is lost to the surroundings and no heat is gained from the surroundings.
- For an adiabatic process, the bulk constant (K) is equal to g times the pressure of the medium (P).

Therefore, the velocity of sound wave in the air is given by

At N.T.P. the density of air is 1.29 kg / m³. And P = 0.76 x 13600 x 9.8 N/m², and ϒ = 1.41

Therefore the velocity of sound in air is given by

v = 332 m/s

- Experimentally the velocity of sound is found to be 332 m/s. the actual velocity of sound in air is almost same as the calculated velocity of sound in air.

**Effect of Temperature on the Velocity of Sound:**

- The velocity of sound in air is given by Laplace’s formula

Where, P = Pressure of medium (air), γ = = Ratio of specific heats of medium (air)

- Consider one mole of atmospheric air having pressure P, volume V, density ρ at temperature T K then we have

PV = RT ………………. (1)

Thus the velocity of sound in air is directly proportional to the absolute temperature of the air.

#### Effect of Change of Pressure on the Velocity of Sound in Air at Constant Temperature:

- The velocity of sound in air is given by Laplace’s formula

Where, P = Pressure of medium (air), γ = = Ratio of specific heats of medium (air)

- Let V
_{1}be the volume of the gas at pressure P_{1}. Let ρ_{1}be the density of a gas at this pressure. Let v_{1}be the velocity of sound under these conditions. Let V_{2}be the volume of the gas at pressure P_{2}. Let ρ_{2}be the density of a gas at this pressure. Let v_{2}be the velocity of sound under these conditions.

Thus there is no effect of the change in pressure on the velocity of sound in air at a constant temperature.

**Effect of Humidity on the Velocity of Sound in Air:**

- The velocity of sound in air is given by Laplace’s formula

Where, P = Pressure of medium (air), γ = = Ratio of specific heats of medium (air)

- Humidity or moisture in the air is due to water vapours present in the medium. The density of vapour is very much less than that of dry air. Hence the density of moist air is less than that of dry air.

Let ρ_{1} be the density of dry air and ρ_{2} be the density of moist air

- Thus the velocity of sound in air is inversely proportional to the square root of the density. Now ρ
_{1}> ρ_{2}

Therefore v_{2}> v_{1}. Thus the velocity of sound in moist air is more than that in dry air.

**To Show that for every 1o C rise in temperature, the velocity of sound increases by about 61 cm/s**

Let v_{o} be the velocity of sound at 0 °C. Let the temperature changes to say t °C. Let v_{t} be the velocity of sound at t °C.

Provided the increase in temperature t is not large. where β = 1/273

As we are finding the increase in velocity of sound per degree rise in temperature t = 1, we get

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