shm and acoustics - drmgriyearbtech.com

UNIT – II

SHM AND ACOUSTICS

2.1. SIMPLE HARMONIC MOTION:

A body is said to execute a harmonic

motion when it moves such that its

acceleration is always directed towards a

certain fixed point and varies directly as its

distance from that fixed point. An

oscillatory or a to and pro motion about a

fixed point with a definite period is called

simple harmonic motion.

Consider a particle P moving with an

uniform speed in a circle of radius ‘a’. As

the particle starting from X completes one

rotation in the direction shown, the foot of

the perpendicular ‘N’ to the diameter YY′moves from O to Y, Y to O, O to Y′ and

back again Y′ to O. The motion of N is said

to be a simple harmonic motion.

Displacement: The distance travelled by a vibrating particle from its mean position

is called the “displacement” of the vibrating body.

Amplitude: The maximum distance travelled by a vibrating particle from its mean

position is called the amplitude of vibrating body. In this case it is equal to O Y or

the radius of the circle (see fig. 2.1).

Period: The time taken by the vibrating particle to complete one vibration is called

the period of the vibrating body. The period of N is the time taken by it to travel

from O to Y, Y to Y′ and back again to O.

Fig. 2.1: Simple harmonic motion

Frequency: The frequency of the vibrating particle is equal to the number of

vibrations made by the particle in one second. It is also equal to the reciprocal of

its period.

Phase: The phase of the vibrating particle at any instant is its state or condition

as regards its position and direction of motion with respect to mean position. The

product ω t gives the phase of the motion where ω is the angular requency and t is

the time.

Wavelength: The wavelength of a wave is defined as the distance between two

successive points which are in the same phase of vibration. It will be therefore the

distance between two successive crests or troughs in a transverse wave. In a

longitudinal wave it is the distance between two successive condensations or

rarefactions.

2.2. EQUATION OF SIMPLE HARMONIC MOTION:

The displacement of the food of the perpendicular from the mean position is

given by

y = O N = O P sin ω t.

Since O P = the radius of the circle = a,

y = a sin ω t,

Velocity = v = dy

dt = a ω cos ω t = ω √a2 − y2

Acceleration = dv

dt =

d2 y

dt2

= − a ω2 sin ωt = − ω2y.

(or) d2 y

dt2 + ω2 y = 0.

This is called the differential equation of free harmonic vibrations. If we solve this

equation we will get y = a sin ωt. Here ‘a’ is a constant called the amplitude. It is

equal to the maximum displacement possible and happens when ωt = 90° = π ⁄ 2.

In a simple harmonic motion, the velocity of the vibrating particle is leading

the displacement by an angle π2

and is lagging behind the acceleration of the particle

by an angle π2

. Further its velocity amplitude (maximum velocity) is a ω and its

acceleration amplitude (maximum acceleration) is a ω2.

Transverse wave motion: Transverse wave motion is that wave motion in which

the particles of the medium vibrate about their mean positions in a direction at right

2.2 ENGINEERING PHYSICS – I

angle to the direction of propagation of the wave. Transverse waves are propagated

in the form of crests (the points having maximum positive displacements) and troughs

(the points having maximum negative displacements). Example: Propagation of light

waves.

Longitudinal wave motion: Longitudinal wave motion is that wave motion in which

the particles of the medium vibrate about their mean positions in the same direction

in which the wave is propagated. The longitudinal waves are propagated in the form

of condensations (the points heaving high pressures) and rarefactions (the points

having low pressures). Example: Propagation of sound waves in air.

2.3. GRAPHICAL REPRESENTATION OF SHM

(Displacement, Velocity and Acceleration Curves):

(i) Displacement Curve:

A graph between the displacement of a particle executing SHM and the time

is called displacement or time-displacement curve. To draw the displacement curve

we use the equation for the displacement of a vibrating particle, that is,

y = a sin ω t but ω = 2 π ⁄ T, where T is the periodic time

y = a sin 2 π t

T

.... (1)

Fig. 2.2

By using eqn. (1), we determine the displacements at different values of time.

The value of y at different values of time are tabulated as:

θ = ω t 0 90° 180° 270° 360°

t 0 T/4 2T/4 3T/4 4T/4

y 0 a 0 − a 0

SHM AND ACOUSTICS 2.3

On plotting r along X-axis and corresponding values of y along Y-axis, the

displacement curve is obtained as shown in fig. 2.2. From this curve it is clear that

the displacement curve of a particle executing SHM is exactly a sine curve.

(ii) Velocity Curve:

A curve showing the variation of velocity of a particle executing SHM with time

is called velocity curve. To draw the velocity curve we use the equation for the

velocity of a vibrating particle, that is, y = a sin ω t

∴ u = dy

dt = a ω cos ω t = a ω cos

2 πT

t.... (2)

Fig. 2.3

With the help of eqn. (2) we can determine the velocity at different values of

time. The value of u at different values of time are tabulated as:

θ = ω t 0 90° 180° 270° 360°

t 0 T/4 2T/4 3T/4 4T/4

u = dy ⁄ dt a ω 0 − a ω 0 a ω

On plotting t along X-axis and corresponding values of u along Y-axis, the

velocity curve is obtained as shown in fig. 2.3. From this curve it is clear that the

velocity curve of a particle executing SHM is exactly.

(iii) Acceleration Curve:

A curve showing the variation of acceleration of a particle executing SHM with

time is called acceleration curve. To draw the acceleration curve we use the equation

for the acceleration of a vibrating particle, that is, y = a sin ω t.

α = d2ky

dt2 = − a ω2

sin ω t = − a ω2 sin

2 πT

t.... (3)


With the help of eqn. (3), we determine the acceleration at different values of

time. The value of α at different values of time are tabulated as:

θ = ω t 0 90° 180° 270° 360°

t 0 T/4 2T/4 3T/4 4T/4

α = d2 y ⁄ dt2 0 − a ω2 0 a ω2 0

Fig. 2.4

On plotting t along X-axis and corresponding values of α along Y-axis, the

acceleration curve is obtained as shown in fig. 2.4. From this curve it is clear that

the acceleration curve of a particle executing SHM is a negative sine curve.

Velocity of Transverse Waves Along a Stretched String:

Let consider the portion PABQ of a string in which a transverse wave is

travelling from left to right with a velocity ν. The string is also drawn towards the

left with the same velocity so that the wave is stationary.

Let AB be a small element of the string. The centre of curvature of the element

AB is O. The angle AOB = θ. It is assumed that the curvature of AB is small and

hence θ is small. Let T be the tension at A and B. The directions of these tensions

are tangential to the element at A and B.

Resolving the tension at A into two rectangular components.

T sin θ2

and T cos θ2

perpendicular and parallel to the string in the undisplaced

portion.

Similarly the components at B will be T sin θ2

and T cos θ2

perpendicular and

paralled to the string in the undisplaced portion.


The parallel components are equal and opposite. The perpendicular components

are along CO.

The resultant tension along CO

= 2T sin θ2

As θ is small, sin θ2

= θ2

Resultant tension = 2T θ2

= T θ .... (1)

For equillibrium position, the resultant tension provide the necessary centripetal

force,

= (m δ x) ν2

R

.... (2)

Therefore (m δ x) v2

R T θ

But θ = δ x

R

Therefore m ⋅ δ x ⋅ ν2

R =

T ⋅ δ x

R

Or ν2 = T

m

v = √T

m

.... (3)

Here m is the mass per unit length of a string.

Also ν = n λ

In case the string of length l vibrates in p segments, length of each segment

= l

p

and each segment corresponds to half wave length.

Therefore l

p =

λ2


or λ = 2l

p

Therefore ν = n (2l)

p.... (4)

Substituting this value of ν in equation (3)

n (2l)p

= √T

m

n = p

2l √T

m

.... (5)

In case of string vibrating in one segment

p = 1

and n = 1

2l √T

m

.... (6)

2.4. ENERGY OF SIMPLE HARMONIC VIBRATIONS:

Let the mass of a body executing simple harmonic motion be m and its

acceleration be d2 y

dt2 when the displacement is y.

Then we have

d2 y

dt2 = − ω2

y

The force required to maintain the displacement y is

m d

2 y

dt2

= − m ω2 y.

The workdone in making a small displacement dy is equal to m ω2 ydy.

This is stored in the particle as its potential energy.

Potential energy V at the displacement y = ∫ 0

y

mω2 y dy = 1

2 mω2 y2.


Kinetic energy at the displacement ‘y’ = 1

2 m

dy

dt

2

= 1

2 m ω2 (a2 − y2).

Total energy of the particle = 1

2 m ω2 y2 +

1

2 m ω2 (a2 − y2) =

1

2 m ω2 a2

Thus the energy of the particle executing simple harmonic motion is directly

proportional to the square of its frequency of vibration.

2.5. FREE AND FORCED VIBRATIONS:

Free Vibrations:

If the body vibrates with its natural frequency which depends upon the

dimensions, moment of inertia and elastic constants of the vibrating system, then

these vibrations are called free vibrations. But if a body is allowed to execute free vibrations

for a length of time, the vibrations die down due to friction and viscous drag.

Example: Vibrations of a tuning fork when it excited and left to itself vibrates with

its natural frequency.

Forced Vibrations:

Forced vibrations are the vibrations of a body when it oscillates with a frequency

different from its natural frequency with the help of a strong periodic force. Thus a

body can be forced to vibrate with any frequency depending upon that of the applied

force but these vibrations die out as soon as the applied force is removed.

Example: An excited tuning fork placed on a table makes the air column below

execute forced oscillations and hence a loud sound is heard due to the greater amplitude

of vibrations of the air column. If the fork is removed no sound will be heard.

Resonance:

Resonance is the phenomenon of setting a body into vibrations with its natural

frequency by the application of a periodic force of the same frequency. Hence even

when the periodic force is removed we can hear the sound. (But in forced vibrations

the body vibrates with frequency different from its natural frequency and we can not

hear the sound when the external periodic force is removed.) Resonance vibrations

are also called “Sympathetic vibrations”.

Example: Resonance of air column using a vibrating tuning fork in resonance of air

column apparatus.

Overtones and Harmonics:

Overtones are the sounds accompanying the fundamental note and bear simple

ratios to the fundamental frequency. Harmonics are the integral multiples of the

fundamental frequency. The Nth harmonic means that its frequency is N times the


fundamental frequency. So the fundamental frequency is called the first harmonic

frequency which is the loudest of all harmonics. The second harmonic frequency is

called the first overtone and so on. The number of overtones present in a sound

determines the quality of sound.

Theoretical analysis:

(a) Free Vibrations:

Let us consider a particle in motion along x-axis with a mass m which is acted

upon a restoring force proportional to the displacement from its equilibrium position.

The restoring force is − s x where s is called force constant or stiffness constant. An

undamped simple harmonic motion of this particle can be written as

m d2 x

dt2 = − sx

∴ d2 x

dt2 =

− sx

m (or) d

2 x

dt2 + ω2 x = 0.

where ω2 =

s

m .

The solution of this equation can be written as

x = a ei ωt + be− i ωt = A sin (ωt + φ)

Here A2 = a2 + b2 and tan φ = a

b

where A is the amplitude and φ is the initial phase of the vibration.

The period of this motion is given by

T = 2 πω

= 2 π √m

s

(b) Damped Vibrations:

Due to friction or air resistance, the free vibrations are damped. So the

amplitude of vibration gradually decreases and after few vibrations, the oscillation

ceases to zero value.

Since the frictional force is proportional to the velocity, this resisting force is

equal to − r dx

dt where r is the damping constant (frictional force per unit velocity).

Thus the equation of motion can be written as

m x.. = − r x

. − sx

m x.. + r x

. + sx = 0


where x.. =

d2 x

dt2 and x

. =

dx

dt

The above equation can be written as

x.. + 2k x

. + ω2x = 0

where 2k = r

m and ω2

= s

m

The solution of this equation can be

written as

x = a e− kt sin √ω2 − k2 t + φ

This equation shows that the amplitude is not constant but diminishes with

time in an exponential manner depending upon the damping constant ‘k’ (fig. 2.5).

Period of damped motion is given by T = 2 π

√ω2 − k

2

Case 1: when k2 < ω2

The displacement slowly decreases and the motion is called under damped and

the ballistic galvanometer follows this kind of motion. So the coil in it comes to

equilibrium position after undergoing few oscillations as shown in figure 2.5.

Case 2: when k2 = ω2

The displacement approaches to zero, much more earlier than in the case 1.

This type of motion is called critically damped. The voltmeter and ammeter pointers

follow this kind of motion.

Case 3: when k2 > ω2

The displacement ‘x’ decays exponentially from maximum to zero. This type of

motion is called overdamped or dead beat and the deatbeat galvanometer follows this

kind of motion.

(c) Forced Vibrations:

Here the motion of the body is governed by the equation

mx.. = − r x

. − s x + F sin pt

where F sin pt is the periodic external driving force.

The above equation can be written as,

mx.. + rx

. + sx = F sin pt.

Fig. 2.5: Damped Vibrations


Put = 2k = r

m , ω2 =

s

m and f =

F

m

x.. + 2k x

. + ω2

x = f sin pt.

The solution can be written as,

x = a sin (pt − θ)

where a = steady amplitude of vibration = f

√(ω2 − p

2)2 + 4k2 p

2

and θ = the initial phase (or) the angle by which the displacement x lags behind the

external force F sin pt and is given by

θ = tan− 1 2kp

[ω2 − p

2]

Case 1: When p < < ω

the amplitude of the forced vibration, a = 1

ω2 = constant and the phase

θ = tan− 1

(0) = 0

Therefore if the frequency of the external force is very much smaller than the

natural frequency of the vibrating particle, the amplitude of the vibrating particle

independent of the frequency of the external force and depends on the magnitude of

the applied force F; The force and displacement are always in phase.

Case 2: When p = ω

a = f

2kp =

F

r ω and θ = tan− 1 (∞) = π ⁄ 2.

Here the amplitude depends on the external force and frictional force. The

displacement lags behind the force by an angle π ⁄ 2.

When p = ω and if there is no frictional force, then the amplitude is maximum

and we get the phenomenon of resonance. In the presence of frictional force, if

p = (ω2 − 2k2)1 ⁄ 2, we can get resonant vibrations.

Case 3: When p > > ω

a −~ f

p2 =

F

mp2

θ = tan− 1 (− 0) = π


In this case the amplitude depends upon the external force frequency and there

is a phase difference of π

Normally in the Case 1 and Case 3 the amplitude of vibration gradually

decreases. If the external force is removed, the vibrations are cut off.

2.6. UNDAMPED VIBRATION:

When a simple harmonic oscillator vibrates with constant amplitude which does

not changes with time, its vibrations are known as undamped vibrations. The energy

of the oscillator executing undamped vibrations remains constant and is independent

time.

2.7. TRANSVERSE VIBRATIONS OF A STRETCHED STRING:

Consider the transverse vibrations

of the stretched string fastened at the

ends. Suppose that the string is initially

distorted into some given curve and

then allowed to swing. The length of the

string is l metre. Initial distortion gives

the displacement such that y = f (x) with

reference to the position of equilibrium

of the string as x axis.

The equation of transverse

vibration is given by

∂2 y

∂ t2

= C2 ∂2 y

∂ x2

where C = √T

m = velocity of transverse vibrations and T = tension on the string

and m = mass per unit length of the string.

Initial conditions:

y = 0 when x = 0

y = 0 when x = l

y = f (x) when t = 0

∂ y

∂ t = 0 when t = 0

Fig. 2.6: Transverse vibrations

along a string


Since y depends on x and t, we write the solution as

y = A exp (α x ± α Ct)

y = A cos α x cos α C t

= A sin α x sin α C t

= B sin α x cos α C t

= B cos α x sin α C t

Let us take one value of y

i.e. y = B sin α x cos α C t

Let α = n πl

∴ y = ∑

n

bn sin

n π x

l cos

n π C t

l

Since y = f (x) when t = 0, put t = 0

∴ y = ∑

n

bn sin

n π x

l

Now f (x) = Σ bn sin n π x

l

∴ bn =

2

l ∫

0

l

f (x) sinn π x

l dx

= 2

l ∫

0

l

f (v) sin n π v

l dv

∴ y = 2

l Σ sin

n π x

l cos

n π C t

l ∫

0

l

f (v) sin n π v

l dv

Since f (v) = b sin n π v

l

∫ 0

l

b sin n π v

l sin

n π v

l dv =

b l

2

∴ y = b sin n π x

l cos

n π C t

l


2.8. LAWS OF VIBRATIONS OF STRETCHED STRINGS:

Let A B be the string fixed at both ends. Suppose this fixed string is plucked

at any point P in the upward direction. Then a vibration will start along PB and

PA. Consider the wave along PA. This wave will be reflected at A, will go to B

where it is again reflected and finally reaches P. At the same time the wave along

PB will be reflected at B, will go to A where it is again reflected and finally reaches

P. If during this time P completes one vibration and is again to start its upward

journey, it will find the incoming waves in phase. The waves thus reinforce and the

vibrations will be maintained. Hence during the time P completes one vibration, the

waves travel a distance,

PA + AB + BP = 2AB = 2 l = λ

where l is the length of the string AB and λ the wave length.

Now, v = n λ (or) n = v

λ

We know that the velocity of the wave along the string, v = √T

m

Hence n = 1

λ √T

m =

1

2 l √T

m.... (2.1)

This is the lowest frequency and the note emitted is called the fundamental. The

string in this case vibrates as a whole in one segment. It is possible that during the

time the waves starting from P travel to A, from A to B and back to P, the particle

P completes 2, 3, .... etc. vibrations. If the string vibrates in p segments, then the

particle P will complete p vibrations during the time the wave travels from P to A,

A to B and back again from B to P. Thus

2 l = p λ

∴ λ = 2 l

p or n =

p

2 l √T

m

Case 1: When the string vibrates in one segment or loop, nodes will be formed at

the ends of the string and the midpoint of the string will have an antinode. Therefore

in this first mode of vibration or fundamental mode of vibration,

l = λ1 ⁄ 2.


Case 2: When the string vibrates in two segments nodes will be formed at the centre

as well as at the ends of the string and each midpoint of the segment will have an

antinode. In this second mode of vibration l = λ2. This second mode of vibration is

called the first overtone or second harmonic.

Case 3: When the string vibrates in three segments we have nodes at the end points

of the segments and antinodes are at the mid points of the segments. In this third

mode of vibration, called the second overtone or third harmonic, l = 3

2 λ3.

The first, second and third harmonic vibrations. Thus the string will vibrate at

different modes of vibrations. From the relation (2.1) we find that the frequency ‘n’

of the fundamental note produced by a string depends upon the following factors:

(i) The frequency is inversely proportional to length of the string. i.e. n ∝ 1

l

(ii) The frequency is directly proportional to the square root of the stretching

force or tension i.e. n ∝ √T.

(iii) The frequency is inversely proportional to the square root of mass per unit

length of the string. i.e. n ∝ 1

√m

The mass per unit length can be written as m = π r2 ρ. Thus it depends upon the

radius of the string and density of the wire. Hence the frequency is inversely

proportional to the radius of the string and inversely proportional to the square root

of the density of the wire.

These three laws are known as the laws of transverse vibrations of strings.

These can be verified experimentally using sonometer.

2.9. SONOMETER AND ITS APPLICATIONS:

Sonometer or monochord consists of a string of steel or brass attached to a peg

P at one end and passing over two fixed knife edges A and B on a wooden sound

box W. The other end of the string passes over a smooth pulley L and carries a

mass M whose weight gives the desired tension to the string. C is a movable knife

edge (or bridge) whose distance from the knife edge B a can be altered when

necessary (fig. 2.7). Usually a metre scale is fixed on the sonometer box to measure

the length of the string between the knife edges.


Fig. 2.7: Sonometer

Verification of I Law: The sonometer wire is suitably loaded say with about 3 to

5 kilogram. A tuning fork of known frequency n1 is excited by striking its prong

gently on a rubber pad and its shank is pressed on the sonometer box. The wide is

plucked between B and C and the two sounds compared. If the sonometer sound is

shriller than of the fork it means that it has a frequency greater than the fork.

Hence movable bridge ‘C’ is moved away from B. If the tuning fork is shriller

the bridge C is moved towards B. This is repeated until the two sounds are equally

shrill. At that position beats (i.e., the waxing and waning of sound) will be heard

which signifies that the two frequencies are slightly different. Slight adjustment of

the bridge C one way or the other now will be sufficient to make the sonometer wire

to vibrate in unison with tuning fork. To check the resonance, a small paper rider

is placed between C and B and the excited tuning fork is pressed on the box. The

wire will have sympathetic vibrations or resonance vibrations when the paper rider

will be thrown away.

The measured length l1 metre between the bridge C and bridge B gives the

length of the string which resonates with the fork, which means that its frequency

is equal to the frequency of the tuning fork n1. Keeping the same tension another

tuning fork of known frequency n2 is chosen and the corresponding length l2 metre

of the same string is found out as before. Then the length l2 of the wire has a

frequency n2.

By the I Law n1 ∝

1

l1

and n2 ∝

1

l2

i.e. n1 ⁄ n2 =

l2

l1 or n1 l1 = n2 l2

.... (2.2)

Keeping the tension and string as constants, the experiment is repeated a

number of times with a number of tuning forks of known frequencies and the product


of frequency and resonating length is shown to be a constant. Meanwhile if we draw

a graph between n and 1/l we can get a straight line passing through origin which

indicates that n l is a constant for a given load or tension and string (fig. 2.8(a)).

The following tabular column could be used to note down the readings.

Tension ‘T’ = mg = newton

Sl. No. Frequency ‘n’

Hz

Resonating,

length ‘l’

m

nl

Verification of II Law:

For proving this law it has to be shown that frequency of a given length L of

the string is proportional to the square root of the tension. But we have no direct

method of finding out the frequency of the string of the given length L. So a tuning

fork of frequency N is taken. A suitable tension T1 is applied to the sonometer string

by loading it with a known mass. Tension = Mg where M is the mass suspended.

As described earlier the resonating length l1 metre of the string for the frequency

N is determined. If n1 is the frequency of string of standard length L under the

same tension T1, then by the I Law,

n1 L = Nl1 (or) n1 = N l1

L

By II Law n1 ∝ √T1 (or) N l1

L ∝ √T1

If the same tuning fork is used throughout the experiment, for a constant length L

of the string,

l1 ∝ √T1 … (a)

The tension is altered to a different value T2 and the resonating length l2 of the

same string for the same fork N is determined.

∴ l2 α √T2 … (b)

Fig. 2.8(a)


By equations (a) and (b), we have

l1

l2 =

√T1

√T2

(or) √T1l1

= √T2

l2

.... (2.3)

This relation can be verified and thus II Law can be proved.

The tension may be varied to different values and the relation √T ⁄ l as a

constant can be proved to be true for a given tuning fork and for the same string.

Further if we draw a graph between √T and l we can get a straight line passing

through origin which proves the constancy of √T ⁄ l and hence II Law is proved

graphically also (fig. 2.8(b)).

Sl. No. Tension T

newton

Resonating

length

m

√Tl

Verification of III Law:

To verify this law different strings are stretched under the same tension. Using

a constant frequency N of a fork, resonating lengths l1 and l2 of the two strings for

the same tension are found out. If n1 and n2 are the frequencies of the two strings

for a standard length L,

n1 = N l1

L and n2 =

N l2

L

i.e. n1 ⁄ n2 = l1 ⁄ l2

By III Law n1

n2 =

√m2

√m1

where m1 and m2 are the linear densities of the two strings.

Fig. 2.8(b)


So we must prove l1

l2 = √m2

m1 (or) l1

⁄ √m1 = l2 √m2 .

.... (2.4)

Strings of different thicknesses and materials could be utilised and it can be

shown that l √m is a constant verifying the III Law. m can be determined knowing

the radius ‘r’ and density of the string ‘ρ’ and using the relation m = π r2 ρ. Draw a

graph between √m and 1 ⁄ l. It will be a straight line which indicates that the truth

of the III Law (fig. 2.6(c)).

Frequency of fork = Hz; Tension = Newtons.

Sl. No. Linear density of

wire ‘m’

kg/m

Resonating

length ‘l’

m

√m l

Determination of frequency of a tuning fork:

First the linear density of the string m is found out by weighing a known length

of the string m =

mass

length or by measuring the radius of the string and knowing

the density m = π r2 ρ . Now the sonometer is set up.

The given tuning fork is sounded and the length of the string that vibrates in

resonance with the fork is determined. The experiment is repeated with different

tensions for the same fork and string and the average value of √T ⁄ l is determined.

Substituting the values of √T ⁄ l and √m in the formula,

n = 1

2 l √T

m =

√Tl

1

2 √m

the unknown frequency ‘n’ can be determined.

Fig. 2.8(c)


Comparison of frequencies of two tuning forks:

Keeping the tension constant, the lengths l1 and l2 of a sonometer string that

vibrates in unison with two given forks under the same tension are found out. If

n1 and n2 are the two frequencies, then by I law.

n1 l1 = n2 l2

(or) n1 ⁄ n2 = l2

⁄ l1

Thus the ratio is found out. If the frequency n2 of one of the forks is known, the

frequency of the other fork n1 can be calculated.

Determination of Specific gravity of a stone:

One can determine the mass of a stone and hence the specific gravity of it using

sonometer. For this we require a tuning fork of known frequency and sonometer.

Instead of known slotted weights, now the string carries the stone whose weight

gives the desired tension to the string. Using a tuning fork of known frequency ‘n’

one can determine the resonating length, l for this unknown tension. So knowing the

values of l, n and m one can determine the value of T from the formula.

n = 1

2 l √T ⁄ m

Since tension = mass × acceleration due to gravity, the mass of the stone can be

calculated. To determine the specific gravity of the stone, first find the mass of the

stone in air and then in water; We know that,

Specific gravity of stone = Weight of the stone in air

Loss of weight in water

= M1

M1 − M2 =

l1 2

l1 2

− l2 2

where M1 and M2 are the masses of stone in air and water respectively; l1 and l2 are

the resonating lengths corresponding to M1 and M2 respectively.

2.10. STANDING WAVE:

In physics, a standing wave − also known as a stationary wave − is a wave

which oscillates in time but whose peak amplitude profile does not move in space.

Faraday observed standing waves on the surface of a liquid in a vibrating container.


2.11. ACOUSTICS:

Acoustics is a branch of physics that deals with the properties of sound waves

and laws of their excitation, propagation and action on obstacles. Acoustic or sound

(sonic) waves are weak disturbances propagated in an elastic medium. These are all

mechanical vibrations of small amplitude. We know that the waves are the

disturbances in the state of matter. Elastic waves or soundwaves are mechanical

disturbances (deformations) propagated in an elastic medium. The propagation of

elastic waves consists of excitation of vibrations of particles of the medium.

2.12. CHARACTERISTICS OF SOUND:

(i) Sound is a form of energy.

(ii) Sound is produced by a vibrating body.

(iii) Sound requires a material medium for its propagation. Sound can be

transmitted through solids, liquids and gases.

(iv) When sound is conveyed through a medium from one point to another point,

there is no bodily motion of the medium.

(v) Sound requires a definite time to travel from one point to another in a

medium. Its velocity is always smaller than the velocity of light.

(vi) Velocity of sound is different in different media. It has the maximum value

in solids, which have higher bulk modulus, and least in gases.

(vii) Sound may be reflected, refracted or scattered. If exhibits diffraction and

interference. In the transverse mode, it exhibits polarisation also.

2.13. CLASSIFICATION OF SOUND:

Sound can be classified into three categories on the basis of frequency. They

are Infrasound, Audible sound, and Ultrasound. Amoung these, Infrasound and

Ultrasound are not audible. Infrasound has the frequencies below 20 Hz. Ultrasound

has the frequencies above 20 kHz.

Further, all the audible sounds can be classified as musical sounds and noises.

Musical Sounds are the sounds which produce pleasing effect on the ear; Musical

sounds are produced when a series of similar impulses follow each other regularly

at equal intervals of time. Noises produce a jarring and unpleasant effect on the ear;

Noises are the sounds of complex nature having an irregular periods and amplitudes.

The characteristics of a musical sound are pitch, loudness and timbre. Among

these, pitch is related to frequency of sound, loudness is related to intensity of sound

and timbre is related to quality of sound.


2.13.1. Pitch:

The pitch of a musical sound is determined by its frequency but it is also a

function of its intensity and wave form. Greater is the frequency of a musical note,

higher is the pitch and vice versa. The frequency and pitch are two different things.

The frequency is a physical quantity and can be measured accurately, while the pitch

of a note is a physiological quantity which is merely the mental sensation experienced

by the observer. The change in pitch with loudness is most pronounced at a frequency

of about 100 Hz. In the 100 Hz range with increasing loudness, the pitch decreases,

eventhough the frequency remains constant. For frequencies between 1000 Hz and

5000 Hz which is the range for which the ear is most sensitive, the pitch of a tone

is relatively independent of its loudness. Generally in the range 20 Hz to 10,000 Hz

the pitch varies in a parabolic manner with frequency.

2.13.2. Loudness:

Loudness measures the amount of sensation produced in the ear and hence

depends upon the listener. Loudness is not a purely physical quantity but is subjective

in nature. Loudness signifies how far and to what extent, sound is audible. Loudness

‘L’ is directly related to intensity ‘I’ and is proportional to log I.

i.e. L = k log I (Weber – Fechner Law)

Here k is a constant depending on the sensitivity of the ear, quality of the

sound and other factors.

Loudness is found to be vary with frequency also. Intensity of sound is a physical

quantity and does not depend upon the listener.

Intensity of sound is measured by the quantity of wave energy flowing across

unit area held normally to the direction of propagation of the sound waves per second

and its unit is watt/m2.

Factors affecting intensity or loudness of sound:

Since I = 2 π2 ν2 a2 v ρ

where ν = frequency, a = amplitude v = velocity of sound and ρ = density of the

medium,

(i) Intensity and hence loudness are directly proportional to the square of the

amplitude.

(ii) Intensity of sound is directly proportional to the density of medium.

(iii) In general, intensity and (hence) loudness vary with frequency.


Ear is more sensitive to frequencies in the middle of the audible range. If

two sounds of frequencies 200 Hz and 2000 Hz be of the same intensity, the

latter appears to be louder to the ear.

(iv) With the increase in the size of a body, larger quantity of air is set into

vibration. So larger the size of a sounding body, greater is the intensity and

louder the sound produced.

(v) A resonant body near the source of sound intensifies the sound. Large

reflectors act as amplifiers; presence of large quantity of absorbing material

reduces loudness.

(vi) The loudness decreases with increase of distance from the source of sound.

Thus loudness varies inversely with the square of the distance from the

source.

(vii) Loudness is greater in the direction of wind motion than in the opposite

direction.

Measurement of Intensity of Sound and Bel:

Intensity is measured in watt/m2. The lowest intensity of sound (at 1000 Hz)

to which a normal human ear can respond is 10− 12 watt ⁄ m2. This is chosen as “Zero”

or “Standard” of intensity. intensity of a sound is measured with reference to this

standard intensity.

The ratio of intensity of a sound to the standard intensity is known as intensity

Level or Relative Intensity of sound. The unit chosen for intensity level is called Bel.

Bel is the intensity level of a sound whose intensity is ten times the standard

intensity. Bel is a large unit and hence decibel (dB) is used in practice.

1 dB = 1 ⁄ 10 bel. Thus

Intensity Level = log (I ⁄ I0) bel = 10 log (I ⁄ Io) dB

where I = intensity of Sound and

I0 = Standard Intensity = 10− 12 watt ⁄ m2 .

1 dB corresponds to log (I ⁄ I0) = 0.1. i.e. (I ⁄ I0

) = 1.26.

Hence 26% change in intensity corresponds to 1 dB of difference in intensity

levels. This is the smallest change is intensity level that a normal ear can detect.

The intensity level is measured in decibel scale (or) logarithmic scale because the

response of human ear to sound is found to be logarithmic.


Table 2.1: Intensity Levels of some Sound Signals

Sound Signal dB Sound signal dB

General noise level 10 Busy traffic 70-80

Whispering 10-20 Running train 100

Passing motor car 30 Passing aeroplane 100

Soft music 40 Thunder 100-110

Average conversation 60-70 Painful Sound > 120

Table 2.1 shows the intensity levels of some sound signals. When the intensity

level is going beyond 120 dB, that sound may give physiological damage to the human

beings.

Since the intensity I = p2

ρ0 c

the Sound Pressure Level (SPL) is also expressed in decibels as

SPL = 20 log (pe ⁄ p0)

where pe = effective pressure of sound and

p0 = reference effective pressure = 0.00002 newton ⁄ m2

The sound level meters are used to measure the intensity level of sounds and

loudness. It consists of a high sensitive microphone of good stability, a linear amplifier

with uniform frequency response, a set of frequency weighting networks and an

indicating meter. The frequency weighting network is to make the readings of the

sound level meter correspond as closely as possible to observed loudness levels. At

high intensity levels, no frequency weighting network is used. Initially, the

microphone converts sound energy into electrical energy in the form of voltage. Then

this voltage is amplified and then passed through a suitable frequency weighting

network and is then used to operate a calibrated indicating meter to read sound

levels in decibels above the standard reference intensity of 10− 12

watt ⁄ m2.


2.14. MEASUREMENT OF LOUDNESS AND PHON:

Loudness level of sound is measured with reference to a standard source

producing a tone of frequency 1000 Hz.

The loudness level (or) equivalent loudness of a sound is said to be one phon,

if the intensity level of standard tone with equal loudness is 1 decibel above the

standard intensity.

As the threshold of audibility varies with frequency, sounds of the same

intensity, but different frequencies, are found to differ in loudness. Conversely, sounds

of different frequencies which appear equally loud to a person are found to have

different intensities. Curves a, b and c represent curves of equal loudness for three

values 100 phon, 50 phon and 0 phon respectively. The intensity corresponding to

point A at frequency 1000 Hz is lesser than the intensity corresponding to point B

at frequency 500 Hz, although the points correspond to same loudness. In view of

this fact, a different unit called Phon is used to measure loudness level or equivalent

loudness.

In measurement, the source of sound whose equivalent loudness has to be

measured, is placed near the standard source in front of the listener. The two sounds

are heard alternatively and the intensity of the standard tone is adjusted such that

it has the same loudness as the sound of the given source. The intensity level of the

standard tone is measured. If the intensity level is N decibels above the standard

intensity, the equivalent loudness is said to be N phons.

The unit of loudness is a sone which is defined as being the loudness of a 1000

Hz tone of 40 dB intensity level. It is also equal to the loudness of any sound having

a loudness level of 40 phons. In the range from 40 phons to 100 phons, empirically

one can write,

log L = 0.033 (LL − 40)

where L = the loudness in sones and

LL = the loudness level in phons.

For a frequency of 1000 Hz. the loudness level LL in phons is by definition

numerically equal to the intensity level in decibels.

∴ LL = 10 log (I ⁄ I0)


Since I0 = 10− 12 watt ⁄ m2

LL = 10 log I + 120

log L = 0.033 (10 log I + 120 − 40)

= 0.33 log I + 2.64

which reduces to

L = 445 I0.33

i.e. L α I1 ⁄ 3 for a 1000 Hz tone.

2.14.1. Timbre:

The timbre of a musical sound is used to distinguish between two tones having

the same intensity level and fundamental frequency but different waveforms. Thus

it expresses our ability to recognize the sound of a violin as different from that of

a trumpet, even when the two instruments are sounding the same note with equal

loudness. Like loudness and pitch, timbre is a complex one. Although it is primarily

dependent on the waveform of the tone being heard, it is also a function of its

intensity and frequency. Normally the timbre or quality of sound depends on the

number of overtones present with the fundamental, their relative intensity and pitch.

We know that open organ pipe produces overtones forming a full harmonic series,

while a closed organ pipe produces only odd harmonics, Hence a note of an open

organ pipe is more pleasant than that of closed organ pipe.

2.15. REVERBERATION:

When a sound pulse is produced in a hall it is reflected into a large number

of times by the objects present in the hall and walls so that a series of waves of

decreasing intensities fall on the listener’s ears. Hence the observer does not hear a

single sharp sound but a “roll of sound”. That is the pulse is heard for a short

definite interval of time. This effect is known as reverberation (or) lingering of sound.

The existence of sound in the hall eventhough the source of sound is cut off is called

reverberation. The time required for the intensity of the sound pulse to fall below

the audible limit is known as the reverberation time of the hall. According to Sabine

the sound produced in a hall gets reflected about 300 times by the various objects

in the hall before it becomes inaudible. Reverberation time is defined as the time

taken by the sound to fall to one millionth of its original value. Taking the velocity

of sound as 330 metre/second, Sabine showed that the time of reverberation

T = 0.0153 V

Σ a S .

where V is the volume of the hall in cubic metre and Σa S is the sum of the products

of absorption coefficients of different surfaces lining the interior of the hall and their


respective surface areas (the absorption coefficient is the ratio of sound energy

absorbed to the sound energy incident). An open window is a cent percent absorber

of sound. In the olden days the palace courts and temple halls were designed and

built to create awe and respect in the minds of the audience. With this aim the halls

were made to have a high reverberation time by having concave ceilings and smooth

walls with small openings. But in modern halls the reverberation is adjusted to be

an optimum value. If it is too much there will be “booming” in the sense that before

the previous sound has died down, the next sound is superposed on it. If it is too

little the sound is “flat” and the performer feels that he has no support. For speeches,

the optimum reverberation time −~ 1 − 2 second and for music it is about 0.5 – 1

second. If the walls are reflecting too much the reverberation will be great. If there

are many openings or if the walls are lined with absorbing materials like felt, the

reverberation will be less. The areas of different portions of the hall are measured

and they are plastered with suitable materials or lined with curtains, to alter the

value of Σ a S suitably. False ceiling is provided to alter the volume of the hall; the

presence of audience in the hall reduces the time of reverberation. Cushioned chairs

are provided, to keep the absorption same whether they are occupied by audience or

not. Thus the reverberation time is adjusted to be equal to the optimum value.

2.15.1. Reverberation Time ‘T’:

Reverberation time ‘T’ is defined as the time taken by sound to fall to one

millionth of its value before the cut off.

Hence taking E = E

m

106 and t = T in equation (iii),

we get eα T = 106

(or) α T = 2.3026 × 6

∴ T = 2.3026 × 6

α =

2.3026 × 6

CA × 4V

Taking the velocity of sound in air ‘C’ = 345 m s− 1

at room temperature, we get

T = 55.2624 V

345 A =

0.16 V

A

Thus T = 0.16 V

Σ a S(iv)


2.16. MEASUREMENT OF ABSORPTION COEFFICIENT:

Since an open window is a perfect absorber of sound, it is taken as a standard

to define the absorption coefficient of a surface. Absorption coefficient of a surface is

defined as the reciprocal of its area which absorbs the same sound energy as absorbed

by an unit area of an open window. For example, 4 square metre of a certain carpet

absorbs the same amount of sound energy as absorbed by 1 square metre of open

window, the coefficient of absorption of that carpet is 1/4. The absorption coefficient

can be measured in terms of the reverberation time. First the reverberation time

T1 is measured when the absorbing material is not in the room. Next the absorbing

material is put inside the room and the reverberation time T2 is measured.

Now 1

T1 =

Σ a S

0.16 V

1

T2

= Σ a S + a

2 S

2

0.16 V

∴ a2 =

0.16 V

S2

1

T2

− 1

T1

Knowing the values of S2 and V, a2 can be calculated.

2.17. FACTORS AFFECTING ACOUSTICS OF BUILDINGS AND THEIR

REMEDIAL MEASURES:

REVERBERATION:

The persistence of sound in the hall even after the source is cut off is called

reverberation. This is due to successive reflections taking place on the walls of the

hall. Too much of reverberation may cause booming sound while too low reverberation

causes flat sound. Therefore the reverberation must be at optimum level for

continuous hearing of sound. This can be obtained by using sound absorbing materials

in walls, ceilings, floor etc.

Reflections and Echoes:

When sound is reflected from walls and other surfaces they produce echoes

which cause a nuisance effect and a change in the original sound. An echo is heard

when the direct and reflected sound waves coming from the same source reach the

listener with a time interval of about 1/7 second. The reflected sound reaching earlier

than this helps in raising the loudness while those arriving later, produces echoes

and cause confusion. These echo effects can be avoided by using sound absorbing

materials and by providing more number of doors and windows.


Echelon effect:

Sometimes a separate musical note with regular phase difference is produced

due to combination of echoes. Thus a set of railings or any regular spacing of

reflecting surfaces may produce a separate musical note due to regular succession of

echoes of the original sound to the listener. This effect is called echelon effect. This

can be avoided if any such regular spacing available is covered with sound absorbing

materials.

Focusing and interference effects:

If there is any concave surface present in the hall, then sound is concentrated

at its focus region and hence dead space at some other region is created. Hence such

surfaces maybe avoided. If present they can be covered with sound absorbing

materials. Similarly if there is interference of direct and reflected waves, interference

patterns create maximum intensities at some places and minimum intensities at some

other places in the hall. All these can be avoided by good sound absorbing materials.

Resonance effect:

Hollows, crevices, window panes etc. select the natural frequencies from the

sound produced in the hall and reinforce them by producing a resonance effect, which

can disturb the original sound. Hence they may produce a jarring effect. These can

be reduced by convex cylindrical segments on the walls and ceilings.

Noises from exterior:

Any noise outside the hall produces a disturbing effect inside. To avoid these,

the opening like window can be covered using screens etc.

2.18. ULTRASONICS:

The human ear can hear the sound waves between 20 Hz to 20 kHz. This range

is known as audible range. The sound waves which have frequencies less than the

audible range are called infrasonic waves.

The sound waves having frequencies above the audible range are known as

ultrasonic waves. Normally they are called high frequency sound waves.

Ultrasonic waves are usually produced by the application of magnetostriction

effect and piezoelectric effect. Galton’s whistle is also a mechanical ultrasonic

transducer.


2.19. MAGNETOSTRICTION ULTRASONIC GENERATOR:

Principle: The change in the dimensions of a ferromagnetic material by the

application of a magnetic field is known as magnetostriction effect. According to this

effect, when a rod of iron or nickel is placed in a magnetic field parallel to its length

a small extension or contraction occurs. This change in length is of the order of one

part in million and is independent of the sign of the field. It depends upon the

magnitude of the field and nature of material. If the rod is placed inside a coil

carrying an alternating current then it suffers the same change in length for each

half cycle of alternating current. This results in setting up vibrations in the rod

whose frequency is twice that of alternating current. On the other hand if the rod

is suitably magnetised before being inserted in the alternating field, then mechanical

change in length will be in step with the alternating current frequency. In such a

case, if the frequency of the alternating current coincides with the natural frequency

of vibration of rod, resonance occurs and the rod vibrates vigourously. Ultrasonic

waves are now emitted from the ends of the rod when the applied current frequency

is of the order of ultrasonic frequency. The frequency of vibration of such a rod is

f = 1

2 l √E

ρ

where l = length of the rod, E = Young’s modulus of the material of the rod material

and ρ = density of the rod. Thus the shorter the rod, the higher sits resonance

frequency. In actual practice one finds it difficult to make a magnotostrictive vibrator

with a very short rod. Since the resonance frequency of long rods is comparatively

small, magnetostrictive oscillators are used to produce low frequency ultrasonic waves.

Pure nickel, nickel alloys (invar, monel metal and perm alloy) and cobalt ferrites are

popular magnetostrictive materials.

Construction:

R is a short permanently magnetised nickel rod which is clamped in the middle

between two knife edges. Coil L2 wound on the right hand portion of the rod along

with a variable capacitor C forms the resonant circuit of the collector tuned oscillator.

Coil L1 wound on the left hand portion of this rod is connected in the base circuit.

The coil L1 is used as a feed-back loop.

Working: When Ebb

is switched on, the circuit L2 C in the collector circuit of

transistor sets up alternating current of frequency f = 1

2 π √L2 C

. This alternating

current flowing round the coil L2 produces an alternating magnetic field of frequency

‘f’ along the length of the rod R. The result is that the rod starts vibrating due to

magnetostrictive effect. The vibrations of the rod create ultrasonics which are sent


out as shown in the figure 2.9. The coil L1 helps in increasing the amplitude of

ultrasonic waves. The longitudinal expansion and contraction of the rod R produces

an e.m.f in the coil L1 which is applied to the base of the transistor which increases

the amplitude of high frequency oscillations in coil L2 due to positive feed-back. By

adjusting the capacitor ‘C’ we can tune the developed alternating current frequency

with the natural frequency of the rod and resonance condition is indicated by the

rise in the collector current shown on the milliammeter.

Advantages:

1. The magnetostrictive generator construction is so simple and its cost is low.

2. At low ultrasonic frequencies, large power output is possible without the risk

of damage of the oscialltory circuit.

Limitations (Drawbacks):

1. It cannot generate ultrasonics of frequency above 3000 k.Hz.

2. The frequency of oscillations depends greatly on temperature.

3. Breadth of the resonance curve is large which is due to variation of elastic

constants of ferromagnetic material with the degree of magnetisation. So we

cannot get a constant single frequency from these magnetostrictive oscillators.

2.20. PIEZOELECTRIC ULTRASONIC GENERATOR:

Principle:

Piezoelectric generator is more efficient than magnetostriction generator.

Therefore modern ultrasonic generators are of this type. It is based on the

piezoelectric effect stated below: If mechanical pressure is applied to one pair of

opposite faces of certain crystals like quartz, cut with their faces perpendicular to

Fig. 2.9: Magnetostrictive ultrasonic generator circuit.


its optic axis, equal and opposite charges appear across its other faces. The sign of

the charges is reversed if the crystal is subjected to a tension instead of pressure.

Thus the sound pressure acting on a crystal microphone develops an a.c. voltage

having the same frequency of the incident sound waves.

But the ultrasonic generation is based on the inverse piezoelectric effect: When

high frequency alternating current is applied to the opposite faces of such a crystal

slice, it expands and contracts periodically. Thus ultrasonic waves are generated. If the

applied alternating current frequency is equal to the natural frequency of the crystal

slice, resonance will occur resulting in a large amplitude of ultrasonic waves.

Preparation of Piezo-electric crystal slice:

Fig. 2.10: Quartz transducer

Natural crystal has the shape of a hexagonal prism with a pyramid attached

to each end (fig. 2.10(a)). The line joining the end points of these pyramids is called

the optic axis or Z-axis. The three lines which pass through the opposite corners of

the crystal constitute its electrical axes or three X-axes and the three lines which

are perpendicular to the sides of the hexagon form it mechanical axes or three Y-axes.

Thin plates cut perpendicular to X axes are called X-cut plates which can generate

longitudinal mode of ultrasonic vibrations whereas those cut perpendicular to Y axes

are known as Y cut plates which can generate transverse mode of ultrasonic vibrations.

Suppose there is a X-cut crystal plate of thickness ‘t’ and length ‘l’. Its thickness is

parallel to X axis, the length is parallel to Y axis and the breadth is parallel to Z axis.

When an alternating voltage is applied along the electrical axis (X axis) of the crystal,

then alternating stresses and strains are set up both in its thickness and length. The

frequency of the thickness vibration is given by


n = p

2t √E

ρ

Similarly the frequency of the length vibration is given by

n = p

2 l √E

ρ

where p = 1, 2, 3 ...... etc. for fundamental, first overtone and second overtone

respectively, E is Young’s modulus and ρ is the density of the material of the crystal

plate.

Both these forms of oscillations are used for generation of ultrasonic waves.

When a quartz is set in length vibrations, sound is emitted from the end surfaces

of the plate as in the case of magnetostriction vibrator. In the case of thickness

vibration, on the other hand, the sound waves are principally emitted in a direction

at right angles to the surface of the plate.

Pieozo electric oscillator circuit:

Fig. 2.11: Piezoelectric oscillator circuit

Construction:

The above circuit is a base tuned oscillator circuit. The coils L2 and L1 of

oscillator circuit are taken from the secondary of a transformer ‘T’. The collect coil

L2 is inductively coupled to base coil L1. The coil L1 and variable capacitor C1 form

the tank circuit of the oscillator.

The crystal plate which is sandwitched between metallic foils forming a parallel

plate capacitor and this is coupled to the electronic oscillator through primary coil

L3 of transformer T.


Working:

When the battery is switched on, the electronic oscillator produces high

frequency oscillations. The oscillation frequency can be varied by the variable

capacitor. By transformer action, an oscillatory e.m.f. is induced in the coil L3. So

the crystal is now under high frequency alternating voltage. Capacitor C1 is varied

till the frequency of oscillation matches with the natural frequency of vibration of

the crystal. Then the crystal will vibrate with large amplitude due to resonance. The

vibrations of the crystal generates high power ultrasonic waves.

Advantages:

1. We can generate ultrasonic frequencies as high as 500 MHz.

2. The breadth of the resonance curve is so small and so we can get a stable

and constant frequency of ultrasonic waves.

3. Using synthetic materials like ceramics A and B, LiNb O3 , Ba TiO3, etc. we

can generate a wide range of frequencies at a lower cost.

4. The piezo-electric ultrasonic generator output is very high and it is insensitive

to temperature and humidity.

Disadvantages:

1. The cost of piezo electric quartz plate is very high and its cutting and shaping

are very complex. But this drawback is eliminated by the use of cheaper

synthetic piezo-electric materials.

2.21. DETECTION OF ULTRASONIC:

Principle:

Whenever there is a change in medium, the ultrasonic waves will be reflected,

is the principle used in ultrasonic flaw detector. Thus from the intensity of the

reflected echoes, the flows are detected without destroying the material. And hence

this method is known as a non-destructive method.

Main components:

A typical ultrasonic flaw detector consists of several functional units such as

1. A master timer.

2. An electronic pulse generator.

3. Single transmitting and receiving transducer.

4. Reflected signal receiving transducer.

5. An echo single amplifier and

6. A display device such as CRO working.


The master timer triggers the signal pulse generator at regular intervals. Signals

generator sends a short burst of high − frequency alternating voltage of the

transducer. The transducer generates pulse of ultrasonic waves. This sound energy

is introduced and propagates through the inspection − piece. When there is a

discontinuity in the wave path. Part of the energy is reflected back from the flaw

surface. The reflected pulse causes the receiving transducer. This induced voltage is

instantaneously amplified and fed to the display device. Initially, when the master

timer triggers the signal pulse generator, simultaneously it activates linear time base

circuit of the display device. Hence in CRO the reflected single strength versus the

time is displayed.

Fig. 2.12: Block diagram for an ultrasonic flaw detector

Medical Applications:

1. Ultrasonics are used to release the contained enzymes.

2. Meniere’s disease produces hearing loss in the ear and it can be cured by

ultrasonic exposure in the ear through the destruction of diseased tissue in

the middle ear.

3. Proliferation: It is used to increase the fibroblasts by stimulation thereby

producing myofibroblasts.

4. Remodelling: It is used to decrease the strength of the scar by affecting the

direction of elastic fibers which are responsible for the scar.

5. Reversible blood cell static: While treating the patient, it is used to gather

the blood in columns separated by plasma.

6. Ultrasonics are also used in cardiology ultrasonic imaging techniques.

Velocity Blood Flow Meter:

Principle:

It works under the principle of doppler effect and doppler shift. (i.e.) There is an

apparent change in frequency of the sound wave s emitted from the source and the

observer, called as doppler effect and the shift in frequency is called as Doppler shift.


Description:

Block Diagram:

Ultrasonic Non-destructive testing:

Principle:

The basic principle behind the ultrasonic inspection is the transmission of the

Ultrasound with the medium and the reflection or scattering at any surface or

internal discontinuity in the medium due to the change in the acoustic impedance.

The Discontinuity means the existence of the flaw or defect or cracks or hole in the

material. The reflected or scattered sound waves are received and amplified and

hence, the defects in the specimen are suitably characterized.

Fig. 2.13

Ultrasonic Imaging System:

Fig. 2.14


Principle of Working:

H During the scanning of the body surface by Ultrasonic transducer, the

Ultrasonic waves are transmitted into the patient’s body.

H The echoes from the body are collected by the receiver circuit.

H Since some echoes come from the depth, they are weak; therefore, proper

depth gain compensation is given by DGC circuit.

H Then these signals are converted into digital signals by an analog to digital

converter and are stored in the memory of the Control Processing Unit (CPU)

of a computer.

H Meanwhile, the control unit in the CPU receives the signals of transducer

position and TV synchronous pulses. These signals generates X plate and Y

plate address information’s for the T.V monitor and is also stored in the

memory of the CPU.

H The stored signals are processed and colour coded and is given to the digital

to analog (D/A converter), which converts the digital signal into analog

signal.

H Finally the mixing circuit mixes the analog signals and TV synchronous

signals properly. The mixed signals are finally fed to the video section of

the television monitor as shown in the figure.

H The TV monitor produces the coloured Ultrasonic image of the internal part

of the Body.

Hazards and safety of Ultrasound:

The safety of exposure to diagnostic ultrasound is evaluated using a structured

approach to risk assessment, based on the acoustic output of present ultrasound

scanners. Thermal hazard is described, the magnitude and probability of temperature

rise is reviewed, and the severity of harm from any outcome is reviewed. Similar

assessments are made separately for acoustic cavitation and gas-body effects, which

have previously been considered together. Finally, radiation pressure is considered in

a similar manner. In each case, means to minimize the risk are suggested where

appropriate. The highest risks are associated with the use of gas-bubble contrast

agents. It is concluded that there is a medium risk associated with trans-cranial

Doppler use, and that this use of ultrasound deserves more detailed safety review.

The risks associated with the current practice of obstetric ultrasound are low. Whilst

the severity of radiation pressure as a hazard is low, it is always present. Little is

known about any associated cell responses and so the associated risk cannot be

evaluated.


2.22. ACOUSTIC GRATING:

Acoustic grating to determine the velocity of ultrasonic waves in a liquid:

Definition:

When ultrasonic wave propagate in a liquid, due to their periodic vibration of

the transducer, pressure varies periodically resulting in density and hence refractive

index variation. This is called acoustical grating. In ultrasonic cell liquid under study

is taken. Ultrasonic transducer is fixed at one side wall inside the cell and ultrasonic

waves are generated.

Diagram:

Fig. 2.15

The waves travelling from the transducer get reflected from the opposite wall

and standing wave pattern is generated. The liquid particles align themselves

according to the wave pattern and they reorganize as regions of nodal points

(maximum density) and antinodal points (minimum densities). Thus the liquid act as

grating in which the loci of nodal points act as opaque region and loci of antinodal

point act as transparent region. Thus acoustic grating is formed. When light falls on

such arrangement diffraction takes place and an image is seen. Applying the theory

of diffraction.

If d is the distance between two adjacent nodes or antinodal planes

d sin θn = n λ


Where n − order of diffraction

λ − wavelength of light

θn − angle of diffraction for nth order.

λa − wavelength of ultrasonic waves

But d = λa

2

Therefore λa

2 sin θn = n λ

λa = 2 n λsin θn

If N is frequency of ultrasonic waves and v is the velocity of the waves in liquid

then

v = N λa

From which velocity of ultrasonic waves can be calculated.

2.23. APPLICATIONS OF ULTRASONICS:

(i) Industrial applications:

Ultrasonic waves are used for detecting cracks or cavities in metal castings.

These are given in chapter on Non destructive testing (Refer chapter 24).

(ii) Ultrasonic welding (cold welding):

The properties of some metals change on heating and therefore they cannot be

welded by electric or gas welding. In such cases the metal sheets are welded together

at room temperature using ultrasonic waves. A hammer is made to vibrate

ultrasonically. The tip of that hammer presses the two metal sheets very rapidly and

the molecules of one metal diffuse into the molecules of the other. Thus the two

sheets get welded without heating.


(iii) Ultrasonic soldering:

Ultrasonic solders are very useful for soldering aluminium foil condensers,

aluminium wires and plates without using any fluxes. An ultrasonic soldering iron

consists of an ultrasonic generator having a tip fixed at its end which can be heated

by an electrical heating element. The tip of the soldering iron melts solder on the

aluminium and the ultrasonic vibrator removes the aluminium oxide layer. The solder

thus gets fastened to the clear metal without any difficulty.

(iv) Ultrasonic drilling and cutting:

Ultrasonics are used for making holes in very hard materials possessing high

impact brittleness such as glass, diamond, gems and ceramics. When ultrasonic

waves pass thorough metals, the phenomenon of cavitation is produced. That is along

the ultrasonic wave path, enormous high pressure air bubbles are produced and that

may collapse within the period of the wave releasing a large amount of pressure and

temperature which are used for cutting and drilling. A suitable drilling tool bit is

fixed at the end of a powerful low frequency (25 kHz) ultrasonic generator. Some

abrasive sherry (a thin paste of carborundum powder and water) is made to flow

between the bit and the plate in which the hole is to be made. Ultrasonic generator

causes the tool bit to move up and down very quickly and the sherry particle below

the bit just remove some material from the plate. This process continues and a hole

is drilled in the plate. The same action is taking place during ultrasonic cutting or

machining.

(v) Ultrasonic cleaning and drying:

Ultrasonic cleaners are extensively employed for cleaning a wide range of articles

like parts of motors, airplanes, machines and electronic assemblies etc. The cleaning

tank is partially filled with cleaning liquid like a water-detergent solution. The

articles to be cleaned are placed in this tank. The generated ultrasonics through the

bottom or side walls of that tank impart high acceleration within the cleaning liquid.

As a result, the molecules of the cleaning liquid get enough kinetic energy. these

high energy molecules strike the dirt particles and remove them away from the

article. The acoustic drying process is appreciably faster than infra red drying. To

remove water from paper during its manufacture, acoustic drying is used.

(vi) Metallurgical use:

Many metals such as iron, lead, aluminium cadmium and zinc etc. which cannot

be alloyed in their liquid state can be brought together when subjected to high

intensity ultrasonic waves.


(vii) Acoustic microscope:

Using high frequency ultrasonic waves (100 MHz) we can study the surface structure

and its homogeneity with higher resolution. Acoustic holograms are used to study the

surface structures of various engineering materials used for space applications.

(b) Medical applications:

(i) Disease treatment:

Ultrasonic therapy has been used to treat diseases such as bursitis, abscesses,

lumbago, etc.

(ii) Surgical use:

Using Ultrasonics, we can remove kidney stones and brain tumors without

shedding blood. Further we can selectively cut any tissue in our body during

an operation.

(iii) Diagnostic use:

Ultrasonics are used for detecting tumors and other defects and locating

abnormal growths in our body. In the case of breast cancers, one can identify

the state of that cancer using ultrasonics in a non destructive manner.

Similarly one can identify the twins or any defect in the growth of fetus

before delivery.

(iv) Dental cutting:

Ultrasonics waves have been found very useful for dental cutting because they

make the cutting almost painless and there won’t be any mechanical device

for cutting.

(v) Ultrasonic doppler blood flow meters are used to study the blood flow

velocities in blood vessels of our body.

(vi) Ultrasonic waves are used for guiding the blind who carries a walking stick

containing ultrasonic transmitter and receiver. Ultrasonic signals reflected

from any obstacle are fed to the head phones through a suitable electronic

circuit which enables the blind person to detect and estimate the distance of

the obstacle.


(c) Ultrasonics as Communication link:

(i) Ultrasonic signalling:

Due to their high frequencies, they can travel many kilometres as a well directed

narrow beam with less absorption through water or air. Hence they have been found

very suitable for direction signalling in submarines and for talking from one vessel

to another. Here high frequency ultrasonic vibrations are modulated by the speech

frequencies in the same manner as in radio transmission.

(ii) Depth sounding and submarine detection:

For depth sounding, the echo sounding technique is adopted. A beam of

ultrasonic waves is directed towards the bottom of the sea from where it is reflected

back to a suitable receiver. The time taken by the waves to go and come back is

recorded accurately. If the velocity of ultrasonic waves in water is known, the sea

depth can be calculated easily. The same method is used for finding the distance

and direction of a submarine, ice-berg or a shoal of fish in the sea. The whole system

is known as SONAR which stands for sound navigation and ranging. It is equivalent

to Radar in air.

(d) Chemical applications:

(i) Ultrasonic waves have been used for forming stable emulsions of immiscible

liquids like water and oil or water and mercury.

(ii) They have been used to improve the homogeneity and stability of photographic

emulsions. Their colour sensitivity has also been increased by the dispersion of the

dye in the emulsion under the influence of ultrasonic waves.

(iii) Proper acceleration is given to supersaturated liquids during their

crystallisation.

(iv) Ultrasonic waves are also used to remove any gas or air bubble in the liquid

metals when they are converted into fused metals.

(v) The coagulation action of ultrasonics finds extensive use in industry for

purification of air.


(vi) The mixtures of different quality can be separated by ultrasonic waves. This

method is used commercially in sorting paper fibers. It facilitates the

manufacture of high quality paper. The long fibers required for thick and

flexible paper may be sorted out from the pulp by low frequency ultrasonic

irradiation.

(e) Physical ultrasonics:

(i) We can study the free volume and adiabatic compressibility of liquids using

ultrasonic waves.

(ii) One can determine the grain size of an alloy or metal from the knowledge

of the attenuation of ultrasonic waves.

(iii) Ultrasonic longitudinal and transverse wave velocities in solids are used to

determine their elastic constants E, N, K and Poisson’s ratio ‘σ’

E = ρ vl 2

N = ρ vs 2

K = ρ vl

2 − 4

3 vs

2

σ = 1

2 1 −

vl

vs

2

− 1

− 1

where vl and vs are longitudinal and transverse wave velocities. Using pulse echo

ultrasonic interferometer, we can determine vl and vs and hence the elastic constants

of the solids or crystals can be evaluated.

(f) Absorption and dispersion of ultrasonic waves:

All media possess absorption and dispersion of ultrasonic waves. Absorption and

dispersion studies are very helpful to understand the molecular structure as well as

the internal structure of the matter in a detailed manner. Dispersion means that the

ultrasonic wave velocity is frequency dependent. Normally dispersion occurs at high

frequencies (107 − 1010 Hz). In solids, the main causes of absorption and dispersion

are scattering from individual grains. In fluids, the main causes are scattering but

more important are the classical causes - viscosity and thermal conductivity and the


non-classical cause - relaxation mechanisms. Relaxation phenomenon arises from the

time - lag inherent in any physical or chemical effect.

The relaxation time ‘τ’ is defined as the time required for a process to proceed

to within 1/e of its equilibrium value.

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