SHM AND ACOUSTICS - drmgriyearbtech.com
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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)
2.4 ENGINEERING PHYSICS – I
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.
SHM AND ACOUSTICS 2.5
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
2.6 ENGINEERING PHYSICS – I
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.
SHM AND ACOUSTICS 2.7
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
2.8 ENGINEERING PHYSICS – I
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
SHM AND ACOUSTICS 2.9
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
2.10 ENGINEERING PHYSICS – I
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
SHM AND ACOUSTICS 2.11
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
2.12 ENGINEERING PHYSICS – I
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
SHM AND ACOUSTICS 2.13
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.
2.14 ENGINEERING PHYSICS – I
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.
SHM AND ACOUSTICS 2.15
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
2.16 ENGINEERING PHYSICS – I
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)
SHM AND ACOUSTICS 2.17
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)
2.18 ENGINEERING PHYSICS – I
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)
SHM AND ACOUSTICS 2.19
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.
2.20 ENGINEERING PHYSICS – I
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.
SHM AND ACOUSTICS 2.21
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.
2.22 ENGINEERING PHYSICS – I
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.
SHM AND ACOUSTICS 2.23
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.
2.24 ENGINEERING PHYSICS – I
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)
SHM AND ACOUSTICS 2.25
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
2.26 ENGINEERING PHYSICS – I
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)
SHM AND ACOUSTICS 2.27
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.
2.28 ENGINEERING PHYSICS – I
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.
SHM AND ACOUSTICS 2.29
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
2.30 ENGINEERING PHYSICS – I
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.
SHM AND ACOUSTICS 2.31
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
2.32 ENGINEERING PHYSICS – I
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.
SHM AND ACOUSTICS 2.33
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.
2.34 ENGINEERING PHYSICS – I
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.
SHM AND ACOUSTICS 2.35
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
2.36 ENGINEERING PHYSICS – I
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.
SHM AND ACOUSTICS 2.37
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
2.38 ENGINEERING PHYSICS – I
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.
SHM AND ACOUSTICS 2.39
(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.
2.40 ENGINEERING PHYSICS – I
(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.
SHM AND ACOUSTICS 2.41
(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.
2.42 ENGINEERING PHYSICS – I
(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
SHM AND ACOUSTICS 2.43
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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2.44 ENGINEERING PHYSICS – I