# What are the characteristics of Sound Waves?

## Introduction to Sound waves

Sound is a form of energy arising due to mechanical vibrations. Hence sound waves require a medium for their propagation. Sound cannot travel in a vacuum. The sound waves are propagated as longitudinal mechanical waves through solids liquids and gases.

## Speed of Sound Waves in Solids, Liquids, Gases

### Newton’s Formula for Speed of Sound Waves

Newton showed that the speed of sound in a medium

$v=\sqrt{\frac{E}{P}}$

E = modulus of elasticity of the medium

P – the density of the medium.

### Speed of Sound Waves in Solids

$v=\sqrt{\frac{y}{P}}$

Y = Young’s modulus of the solid

P = density of the solid

### Speed of Sound Waves in Liquid

$v=\sqrt{\frac{B}{P}}$

B – Bulk modulus of the liquid

P – Density of the liquid

### Speed of Sound Waves in Gases

Newton considered the propagation of sound waves through gases as an isothermal process. Absorption and release of heat during compression and rarefaction will be balanced, thus, the temperature remains constant throughout the process. Then he gave the expression for velocity of sound in air as

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

P  is the pressure of the gas (1.1013×105 N/m2)

ρ is the density of the air (1.293 kg/m3)

On substituting the value of pressure and density the speed of sound obtained was 280 m/s.

There was a huge discrepancy in the speed of sound determined by using this formula with the experimentally determined values. Hence a correction to this formula was given by Laplace it is known as Laplace correction.

## Laplace Correction

According to Laplace, the propagation of sound waves in gas takes place adiabatically. So the adiabatic bulk modulus of the gas (γP) has to be used hence the speed of sound waves in the gas:

$V=\sqrt{\frac{\gamma P}{\rho}}$

γP – adiabatic bulk modulus of the gas

ρ – the density of the medium

For air, γ = 1.41

Substituting the values the speed of sound value obtained was 331.6 m/s.

The values obtained by Newton – Laplace formula is in excellent agreement with the experiment results.

## Factors Affecting the Speed of Sound in Gases

• Effect of pressure
• Effect of temperature
• Effect of density of the gas
• Effect of humidity
• Effect of wind
• Effect of change in frequency (or) wavelength of the sound wave
• Effect of amplitude

### Effect of Pressure

If the pressure is increased at a constant temperature then according to the equation of state PV = RT. If M is the molecular weight and ρ is the density of the gas, then V = M/ρ.

Then we have

P(M/ρ) = RT

P/ρ = RT/M

At constant temperature, if pressure changes then the density also changes in such a way that

P/ρ = constant

So change in pressure does not affect the speed of sound waves through a gas at constant temperature.

### Effect of Temperature

Velocity of sound in a gas

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

But PV = RT for a gas and P = RT/V

$v=\sqrt{\frac{\gamma }{\rho }.\frac{RT}{V}}$

v∝√T

Therefore, the speed of sound is directly proportional to the square root of its absolute temperature.

### Effect of Density

From the velocity of sound in the gas

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

The speed of sound is inversely proportional to the square root of the density of the gas.

### Effect of Humidity

The density of water vapour is less than that of dry air. The presence of moisture decreases the effective density of air hence the sound wave travels faster in moist air or humid air than in dry air.

### Effect of wind

Wind simply adds its velocity vectorially to that of the sound wave if the component of Vw of wind speed is in the direction of the sound wave, the resultant speed of sound is

V resultant = V + Vw

Vw – wind speed

### Effect of Change in Frequency (or) Wavelength of the Sound Wave

Change of frequency (or) wavelength does not affect the speed of sound in a medium (Homogeneous isotropic medium). Sound travels at the same speed in all directions.

V = λf= constant

When the sound wave passes from one medium to another medium, the frequency remains constant but wavelength and velocity changes.

### Effect of Amplitude

From velocity relation

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

Generally, the small amplitude does not affect the speed of sound in the gas. However, a very large amplitude may affect the speed of the sound wave.

## Relation between Speed of Sound in Gas and RMS Speed of Gas Molecules

From velocity of sound wave

$V\sqrt{\frac{\gamma P}{P}}=\sqrt{\frac{\gamma PV}{M}}$

pv = nRT

n = 1

PV = RT

$V=\sqrt{\frac{\gamma RT}{M}}$ then rms speed of gas molecules,

${{V}_{rms}}=\sqrt{\frac{3RT}{M}}=\sqrt{\frac{3}{P}}\sqrt{\frac{\gamma RT}{M}}=\sqrt{\frac{3}{P}}\,\,V$

${{V}_{rms}}=\sqrt{\frac{3}{P}}\,\,V$

Where, V – is speed of sound waves through gas.