# Mechanical Wave

## A complete note on Mechanical Wave useful for NEB students. This note is provided by tutors of different college and instutions.

Mechanical Wave

The wave that requires a medium for it's propagation is called mechanical wave. Example: Sound wave. The velocity of a sound wave in a medium depends upon elasticity of the medium as well as density of the medium.

If 'v' be the velocity of sound wave in a medium having elasticity 'E' and density 'ρ' then,
$$v \propto E^{a} \rho^{b}$$
where a and b are constant whose value is to be determined.

$$or, v = k E^{a} \rho^{b} ............(i)$$
where k is proportionality constant which have no dimension.

Using corresponding dimension formula,

$$[LT]^{-1} = [ML^{-1}T^{-2}]^{a} [ML^{-3}]^{b}$$
$$[LT^{-1}] = [M^{a+b} L^{-a-3b} T^{-2a}]$$

Equating corresponding powers on both side we get,

a+b = 0.............(ii)
-a-3b = 1 ............(iii)
-2a = -1 ............(iv)

Now,

-2a = -1
or, a = 1/2

Also,

$$\frac{1}{2} + b = 0$$
or, 1 + 2b = 0
or, b = - 1\2

Putting the value of a and b in equation (i) we get,

$$v = k E^{\frac{1}{2}} \rho^{\frac{-1}{2}}$$
$$v = k \sqrt{\frac{E}{\rho}}$$

In SI Unit , k = 1
$$v = \sqrt{\frac{E}{\rho}}$$

Velocity of sound in air (Newton's Formula)

The velocity of sound in air is,
$$v = \sqrt{\frac{E}{\rho}}$$
where, B = Bulk modulus of elasticity

ρ = density of air

Sound wave is a longitudinal wave that propagates in the form of compression and rare faction. Newton assume that during propagation of sound wave in air pressure as well as volume of a medium changes continuously but temperature remains constant. So, the process is iso thermal .

For iso thermal process,
PV = constant

Diff. on both sides we get,
P dV + V dP = 0
P dV = - V dP
$$P = \frac{-dP}{\frac{dV}{V}}$$

(-ve) sign indicates that when pressure increases , volume of medium decreases.

$$P = \frac{Stress}{Strain} = B$$

Replacing B by P in equation (i) we get,

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

This is Newton's formula for velocity of sound in air.

At N.T.P , using value of 'P' and 'ρ' in equation (ii) we get,

V ≈ 280 m/s.

But, the experimentally determined value of velocity of sound in air is 332 m/s. Hence, Newton's formula needs a correction.

Laplace's Correction

Laplace has assumed that during the propagation of sound wave in air pressure, volume as well as temperature of the medium changes continuously. This is the adibatic process. For this process we must have,

$$PV^{\gamma} = constant$$
where, $$\gamma = \frac{C_{p}}{C_{v}}$$

Diff. on both sides we get,

$$P \gamma V^{\gamma - 1} dV + V^{\gamma} dP = 0$$
$$\gamma PV^{\gamma - 1} dV = - V^{\gamma} dP$$
$$\gamma P = \frac{- V^{\gamma} dP}{V^{\gamma - 1} dV}$$
$$\gamma P = \frac{\frac{- dP}{V^{\gamma - 1} dV}}{V^{\gamma}}$$
$$\gamma P = \frac{- dP}{\frac{dV}{V}} = \frac{Stress}{Strain} = B$$

Replacing 'B' by \gamma P in equation (i),
$$v = \sqrt{\frac{\gamma P}{\rho}}$$

This is Lapalace's formula to calculate velocity of sound in air.

At N.T.P, using value of \gamma = 1.4 (for air) ,
v ≈ 332 m/s

Hence, Laplace formula gives correct measurement for the speed of sound in air.

Factors Affecting velocity of sound in air

1. Effect of temperature
The velocity of sound in air is,
$$v = \sqrt{\frac{\gamma P}{\rho}}$$
$$v = \sqrt{\frac{\gamma P}{\frac{M}{V}}}$$
where M= molar mass of a gas
$$v = \sqrt{\frac{\gamma P V}{M}}$$
Using ideal gas equation, PV=RT(for 1 mole)
Then,
$$v = \sqrt{\frac{\gamma R T}{M}}$$
Since, R , \gamma and M are constant.
$$v \propto \sqrt{T}$$
For gas at two different temperature T1 and T2 ,
$$\frac{v_{1}}{v_{1}} = \sqrt{\frac{T_{1}}{T_{2}}}$$
2. Effect of pressure
We have,
$$v = \sqrt{\frac{\gamma P}{\rho}}$$
At constant temperature,
When pressure increases, volume of a medium decreases and corresponding density of medium will increase. Hence, the ratio $$\frac{P}{\rho}= constant$$
Since,
γ itself is a constant. So there is no effect of pressure to the velocity at constant temperature.
3. Effect of density
Consider two gases of same nature (same value of
γ) at constant pressure with densities ρ1 and ρ2. If v1 and v2 be the velocity of sound in corresponding gases,
$$v_{1} = \sqrt{\frac{\gamma P}{\rho_{1}}}$$
$$v_{2} = \sqrt{\frac{\gamma P}{\rho_{2}}}$$
Dividing the above equations we get,
$$\frac{v_{1}}{v_{2}} = \frac{\rho_{2}}{\rho_{1}}$$
i.e. $$v \propto \frac{1}{\rho}$$
4. Effect of humidity
The presence of humidity in air reduces it's density. The density of humid air is less than density of dry air.
Since, $$v \propto \frac{1}{\rho}$$
Hence, sound travels faster in humid air than in dry air.
5. Effect of wind

Here, v = velocity of sound along horizontal
v' = resultant velocity of sound along horizontal
vw = velocity of wind in a direction making an angle \theta with horizontal
Then, v' = v + vw cos
θ
For θ = 0
v' = v + vw (max)
For
θ = 90
v' = v (constant)
For
θ = 180
v' = v - vw (min)

Special Thanks(Credits) : Hari Saran Regmi Sir