# Sound Waves

## Introduction to Sound waves

Sound is a form of energy arising due to mechanical vibrations. Hence sound waves require a medium for its propagation sound cannot travel a medium for its propagation. Sound cannot travel in 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 PV = constant (as the medium is into getting heated up when sound is passing through it.) then he stated

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

P – Pressure of the gas (isothermal Bulk modulus of gas) 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:

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

γP – adiabatic bulk modulus of the gas

P – the density of the medium

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 by Boyle’s law

PV = constant [for a fixed mass of gas]

P = density of the gas (for the fixed value of density)

$\frac{P}{P}=constant$

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

### Effect of Temperature

Velocity of sound in a gas

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

$P=\frac{M}{V}$

∴ $\,\,\,\,\,V=\sqrt{\frac{\gamma PV}{M}}$

For perfect gas

PV = nRT [for 1 mole of gas]

$V=\sqrt{\frac{\gamma RT}{M}}$

$V\alpha \sqrt{T}$

So,

$\frac{{{V}_{1}}}{{{V}_{2}}}=\sqrt{\frac{{{T}_{1}}}{{{T}_{2}}}}$

### How Density Affects the Speed of Sound?

From the velocity of sound in the gas

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

$V\alpha \frac{1}{\sqrt{P}}$

### Effect of Humidity

Under the same conditions of temperature and pressure, the density of water vapor is less that of dry air in the presence of moisture decreases the effective density of air hence the sound wave travels faster in moist 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 do not affect the speed of sound in a medium (Homogeneous isotropic medium) Sound travels with the same speed in all directions.

V = λF = constant

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

### Effect of Amplitude

From velocity relation

$V\sqrt{\frac{\gamma P}{P}}$ (for some amplitudes)

Generally, 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.