Mechanical Wave

Mechanical waves  are in  form of disturbance  that travels  through the medium due to the periodic  motion of the particles about their mean position, the  disturbance being transferred  from  one  particle to another particle.eg   sound  waves,  vibration on the string . The medium through which waves propagate should have the property of   volume elasticity .Not all the waves are mechanical in nature. Waves like light waves, radio waves, x-rays   do not need material medium to propagate.

Velocity of sound in solid and liquid

Velocity of longitudinal wave in solid: Velocity of longitudinal wave depends on the modulus of elasticity, which depends on the shapes and size of the solid.

Velocity of longitudinal wave in thin rod:  v = Yρ−−√$\sqrt{\frac{\mathrm{Y}}{\rho }}$   where Y = young modulus of elasticity And ρ is density of medium (rod)

Velocity of longitudinal wave in solid of other shape v = K+43ηρ−−−−−√$\sqrt{\frac{\mathrm{K}+\frac{4}{3}\eta }{\rho }}$  where, K= bulk modulus of elasticity, η = modulus of rigidity

Velocity of longitudinal wave in liquids, v =  Kρ−−√$\sqrt{\frac{\mathrm{K}}{\rho }}$   where K= bulk modulus of elasticity.

Velocity of longitudinal wave in gases   v = Kρ−−√$\sqrt{\frac{\mathrm{K}}{\rho }}$   where K= bulk modulus of elasticity.

 Velocity of transverse wave

 V = T√√m$\frac{\sqrt{\mathrm{T}}}{\surd \mathrm{m}}$   where T= tension applied to the string  and m is the mass per unit length .

Velocity of sound in Gas (air)

Newton’s formula

We know that the properties of a medium that governs the propagation of a mechanical wave are:

a. Restoring force

b. Inertial mass

The restoring force acting on the particles of the medium is intimately connected to the approximate elastic modulus of the medium and the inertial mass, to its density.

 Newton derived an expression for the velocity of sound in a homogenous medium is

V=Eρ−−√$\sqrt{\frac{\mathrm{E}}{\rho }}$………………..1

Where V is the velocity of sound, E is the modulus of elasticity and ρ is the density of the medium. Let’s say that if the medium is a gas then we only consider the bulk modulus and give the relation.

V=Bρ−−√$\sqrt{\frac{\mathrm{B}}{\rho }}$………………….2

Where B is the bulk modulus of elasticity. Newton assumed that when sound propagates through the medium, compression and rare faction occurs and energy exchanged   with surroundings take place in such a slow manner that temperature of the medium remains the same. i.e sound propagates through the isothermal process and applied  Boyle's law.

 At a region of compression, the pressure increases and volume decreases.

Let the initial pressure and initial volume are P and V , and the final pressure and volume are P+dP and V-dV. Here dP is increase in pressure and dV is decrease in volume at the region of compression.

Applying Boyle's law, (P + dP) (V -dV) = PV

Or,  PV - dPV + dVP - dP.dV = PV…………..3

Since the changes in pressure and volume are small, dP.dV can be neglected. Then, from 3weget

- dVP + dPV =0

P=VdPdV$\mathrm{V}\frac{\mathrm{d}\mathrm{P}}{\mathrm{d}\mathrm{V}}$…………..4

By the definition of the bulk we have B=changeinpressurechangeinvolumeoriginalvolume$\frac{\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{e}\mathrm{i}\mathrm{n}\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{e}}{\frac{\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{e}\mathrm{i}\mathrm{n}\mathrm{v}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{m}\mathrm{e}}{\mathrm{o}\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{v}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{m}\mathrm{e}}}$

Therefore we have B=dPdVV$\frac{\mathrm{d}\mathrm{P}}{\frac{\mathrm{d}\mathrm{V}}{\mathrm{V}}}$

Or, B=VdPdV$\mathrm{V}\frac{\mathrm{d}\mathrm{P}}{\mathrm{d}\mathrm{V}}$………………..5

Therefore from eq. 4 and 5 we find that P=B

Therefore, Newton's formula for velocity of sound can be written as

V=Pρ−−√$\mathrm{V}=\sqrt{\frac{\mathrm{P}}{\rho }}$

At NTP the pressure of air P = 0.76 x 9.8 x 13.6 x 103Nm-2

= 1.013 x 105 Pa or Nm-2

Therefore V=1.013∗1051.293−−−−−−−√=280m$\sqrt{\frac{1.013\ast {10}^5}{1.293}}=280\mathrm{m}$/s

But the theoretical value velocity of sound at 00Cis 332ms-1.Thus Boyle's law does not apply in this case.

Later, Laplace corrected the Newton’s formula for the velocity of sound by assuming that the process is adiabatic.

Laplace Correction

Laplace corrected Newtons   formula by assuming that  process of compression and rarefaction  occurs so rapidly that  there is no sufficient time to  exchange  heat  energy with surroundings so the process is not isothermal  but it is adiabatic as the total quantity of the heat  of the system remain constant.

The relation between pressure and volume of a gas under adiabatic conditions is given by

PVγ${\mathrm{V}}^{\gamma }$=a constant.

We have γ =CPCV$\frac{{\mathrm{C}}_{\mathrm{P}}}{{\mathrm{C}}_{\mathrm{V}}}$

Let the pressure change by an amount dP, producing a change in volume by dV. Then

PVγ${\mathrm{V}}^{\gamma }$=(P+dP) (V-dV)γ

Taking out Vγ${\mathrm{V}}^{\gamma }$from the second factor from the above expession

PVγ${\mathrm{V}}^{\gamma }$=(P+dP).Vγ(1−dVV)γ${\left(1-\frac{\mathrm{d}\mathrm{V}}{\mathrm{V}}\right)}^{\gamma }$…………………..6

But from the binomial expansion

(1−dVV)γ≈1−γdVV${\left(1-\frac{\mathrm{d}\mathrm{V}}{\mathrm{V}}\right)}^{\gamma }\approx 1-\gamma \frac{\mathrm{d}\mathrm{V}}{\mathrm{V}}$

Now from equation 6 we have

P=(P+dP). (1−γdVV)${\left(1-\gamma \frac{\mathrm{d}\mathrm{V}}{\mathrm{V}}\right)}^{}$…………………..7

P=P- γPΔVV$\frac{\mathrm{\Delta }\mathrm{V}}{\mathrm{V}}$ + dP-γV.dP.dV$\frac{\gamma }{\mathrm{V}}.\mathrm{d}\mathrm{P}.\mathrm{d}\mathrm{V}$

Canceling P on both sides and neglecting the term containing dP.dV because it is too small, we get

-γPdVV$\frac{\mathrm{d}\mathrm{V}}{\mathrm{V}}$ + dP=0

γPdVV$\frac{\mathrm{d}\mathrm{V}}{\mathrm{V}}$ = dP

γP=VdV$\gamma \mathrm{P}=\frac{\mathrm{V}}{\mathrm{d}\mathrm{V}}$ .dP

But the LHS in the above equation represents the bulk modulus,

B= γP

From eq. 2 we have

V=γPρ−−−√$\sqrt{\frac{\gamma \mathrm{P}}{\rho }}$

This is known as Newton-Laplace formula for the velocity of sound in a gas.

For air γ$\gamma$=1.4,now substituting the value of γ$\gamma$,P&ρ$\rho$ we get  the velocity of sound in the air approximately 331.6m/s which is good agreement with  experimental value.

 Factors affecting the velocity of the sound in air

Pressure: There is no effect of the pressure on velocity of sound because the ratio of P/ρ is constant i.e. when the pressure increases, density also increases, and if pressure decreases, then density decreases at constant temperature.  Since we have the relation v=γPρ−−−√$\sqrt{\frac{\gamma \mathrm{P}}{\rho }}$  where γ is constant   and P/ρ is also constant at a given temperature. From the above relation we have v=constant.

Temperature:  since we have a relation v=γPρ−−−√$\sqrt{\frac{\gamma \mathrm{P}}{\rho }}$  we also have ρ=M/V where M be the mass of the gas V be the volume. Then v=γPVM−−−−√$\sqrt{\frac{\gamma \mathrm{P}\mathrm{V}}{\mathrm{M}}}$ ………..1

Now from ideal gas equation for one mole gas we have relation PV=RT…..2

From 1 and 2 we get

v=γRTM−−−−√$\sqrt{\frac{\gamma \mathrm{R}\mathrm{T}}{\mathrm{M}}}$ since γ, R and ρ is constant then we have v=CT√……….3$\mathrm{C}\sqrt{\mathrm{T}}\dots \dots \dots .3$where C=γRM−−−√$\sqrt{\frac{\gamma \mathrm{R}}{\mathrm{M}}}$

now from equation 3 we have relation that velocity of the sound is proportional to the square root of the absolute temperature at given pressure.

Density: since we have the relation v=γPρ−−−√$\sqrt{\frac{\gamma \mathrm{P}}{\rho }}$  let us consider two gases at the same temperature then we have two relation i.e.

v1=γPρ1−−−√………….1${\mathrm{v}}_1=\sqrt{\frac{\gamma \mathrm{P}}{{\rho }_1}}\dots \dots \dots \dots .1$ for one gas and

v2=γPρ2−−−√………….2${\mathrm{v}}_2=\sqrt{\frac{\gamma \mathrm{P}}{{\rho }_2}}\dots \dots \dots \dots .2$ for another gas.

from$\mathrm{f}\mathrm{r}\mathrm{o}\mathrm{m}$ Equation 1 and 2 we have

v1v2=ρ2ρ1−−√$\frac{{\mathrm{v}}_1}{{\mathrm{v}}_2}=\sqrt{\frac{{\rho }_2}{{\rho }_1}}$  in general we have v α 1ρ√$\frac{1}{\sqrt{\rho }}$                                 

This is the relation of the velocity of the sound and the density.

This relation show that the gas which has lighter density has higher velocity and those have higher density has lower velocity

Humidity or moisture:  Since the density of moisture gas is less than the density of dry gas and we have relation v α 1ρ√$\frac{1}{\sqrt{\rho }}$ due to this relation the velocity of moisture is greater than that of dry air.

Frequency or wavelength:  As we have relation v=γPρ−−−√$\sqrt{\frac{\gamma \mathrm{P}}{\rho }}$ in this relation, frequency and wavelength has no effect in the formula of velocity of sound so that the sound of any frequency or wavelength travels through a given material with the same velocity.