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### Velocity of wave

#### (i) Velocity of a longitudinal wave in liquid and gas

Where,

β = bulk modulus of elasticity of the medium.

ρ =density of the liquid or gas

#### (ii) Velocity of a longitudinal wave in solid

Where,

Ү = Young modulus of elasticity.

ρ =density of solid.

#### The velocity of longitudinal wave on a stretched string

Where,

T = Tension on the string

µ=mass per unit length of string

#### The velocity of electromagnetic wave

Where,

E = Permeability of medium

µ=permeability

### The velocity of sound in any medium by the Dimensional method

A sound wave is a longitudinal wave. The velocity ‘v’ of a sound wave in any medium depends upon the medium of elasticity ‘E’ and density of medium ‘ρ’ through which it travels

The dimensional equation of equation (i) is,

Equating dimensions on both sides,

a + b = 0 — (ii)

-a-3b = 1 — (iii)

-2a = -1 — (iv)

Solving these equations we get,

using the value of a and b in equation (i)

Here, k = 1

for solid E = γ (Young modulus)

for liquid and gas E = b (Bulk modulus)

#### Newton’s formula for the velocity of sound

According to newtons when a sound wave travels through a medium, the volume and pressure can be changed but the temperature of the medium remains constant. Hence it obeys the condition of the isothermal process. The equation of state for an isothermal process is,

Pv= constant—(i)

Differentiating equation (i) on both sides

(where ß = Bulk modulus of air )

This is newtons formula for the velocity of sound in air.

At NTP

p = 1.01 x 10^5 N/m^2

þ = 1.293 kg/m^3

∴ The velocity of sound at NTP

v = √1.01 x 10^5 /1.293

= 280 m/s

This value does not match the experiment value. Therefore Laplace corrected newtons formula by suggesting that during the propagation of sound in air temperature does not remain constant.

### Laplace Correction

According to Laplace, when a sound wave travels through the air at the compression region temperature increases, and at the rarefaction temperature decreases the propagation of the sound wave. Therefore propagation of sound waves takes place under an adiabatic process.

The equation of state for the adiabatic process is

PVˆr = Constant

Differentiating the above equation on both sides

So, the Velocity of sound in air,

This is the Laplace formula

At STP

This results closed agree with the experimental value. Thus Laplace formula gives the correct values of velocity of the sound in air.

### Factors that affect the velocity of sound

#### 1. Effect of density of gas

The velocity of sound in air is given by

From the above formula for constant pressure

This shows that at constant pressure velocity of sound is inversely proportional to the square root of density of gas.

#### 2. Effect of temperature

For given gas at one mole,

PV = RT

### Effect of pressure

We have,

This means the velocity of sound is independent of pressure.

#### 4. Nature of gas

We have,

This shows that the velocity of sound in a gas is inversely propositional to the square root of the molecular mass of gas. Hence, the velocity of sound in hydrogen gas is greater

#### 5. Effect of humidity

As the increase in humidity in a gas its density decreases but we have a relationship.

This shows that as the decreases in density its velocity increases. Hence velocity of sound increases when humidity increases.

#### 6. Effect of Wind

Let ‘Vs’ and ‘Vw’ be the velocity of sound and wind respectively and the angle between them is ‘θ ‘ as shown in figure

Now, the resultant velocity of sound alone OA

Case I

When θ = 0°

i.e The velocity of wind and sound is in the same direction.

In this case velocity of sound increases.

Case II

When θ = 180°

i.e The velocity of wind and sound are in opposite directions.

In this case velocity of sound decreases.

Case III

When θ = 90°

i.e The velocity of wind and sound are Perpendicular.

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