MAHARASHTRA STATE BOARD
CLASS XI and XII
CHAPTER 6
SUPERPOSITION OF WAVES
CONTENT:
Introduction
Transverse and
longitudinal waves
Displacement relation in a
progressive wave
The speed of a travelling
wave
The principle of
superposition of waves
Reflection of waves
Beats
Doppler effect
TataKelola dan KamSiber Kecerdasan Buatan v022.pdf
ย
Chapter 6 - Superposition of waves.pptx
1. Std : 12th Year : 2022-23
Subject : PHYSICS
Chapter 6 : Superposition of waves
CLASSXII
MAHARASHTRA STATE BOARD
2. Can yourecall ?
1. What is wave motion?
2. What is a wave pulse?
3. What are common properties of waves?
4. What happens when a wave propagates?
5. What are mechanical waves?
6. What are electromagnetic waves?
7. How are mechanical waves different from
electromagnetic waves?
8. What are sound waves?
3. 1. What is wave motion?
Ans: A wave is a periodic or oscillatory disturbances in a medium
which is propagated from one place to another with a finite
speed.
2. What is a wave pulse?
Ans: A disturbance caused locally for a short duration of time is
called as wave pulse.
3. What are common properties of waves?
Ans: a) Period (T)
b) Frequency (n)
c) Velocity (v)
d) Amplitude (A)
e) Wavelength (๐)
4. 4. What happens when a wave propagates?
Ans: When a wave propagates, it produces disturbance or
oscillation in the medium which results into transfer of
energy.
5. What are mechanical waves?
Ans: A mechanical wave is a disturbance produced in an elastic
medium due to periodic vibrations of particles of the medium
about their respective mean position.
6. What are electromagnetic waves?
Ans: Wave which do not require material medium for their
propagation are called electromagnetic waves.
5. 7. How are mechanical waves different from electromagnetic
waves?
Ans: Mechanical waves
requires material
medium for their
propagation whereas
electromagnetic waves
do not require any
material medium for
their propagation.
6. 8. What are sound waves?
Ans: Sound wave is a pressure wave consisting of compressions
and refractions travelling along the direction of
propagation of the wave.
Sound wave are longitudinal wave which exhibit various
phenomena like echo, reverberation and doppler effect of
waves.
7. Progressive Wave
A wave, in which the disturbance produced in the medium travels in a
given direction continuously, without any damping and obstruction, from
one particle to another, is a progressive wave or a travelling wave.
Properties of progressivewaves
1) Each particle in a medium executes the same type of vibration. Particles vibrate
about their mean positions performing simple harmonic motion.
2) All vibrating particles of the medium have the same amplitude, period and
frequency.
3) The phase, (i.e., state of vibration of a particle), changes from one particle to
another.
4) No particle remains permanently at rest. Each particle comes to rest
momentarily while at the extreme positions of vibration.
5) The particles attain maximum velocity when they pass through their mean
positions.
8. 6) During the propagation of wave, energy is transferred along the wave.
There is no transfer of matter.
7) The wave propagates through the medium with a certain velocity. This
velocity depends upon properties of the medium.
8) Progressive waves are of two types - transverse waves and longitudinal
waves.
9) In a transverse wave, vibrations of particles are
perpendicular to the direction of propagation of
wave and produce crests and troughs in their
medium of travel. In longitudinal wave,
vibrations of particles produce compressions and
rarefactions along the direction of propagation
of the wave.
10) Both, the transverse as well as the longitudinal,
mechanical waves can propagate through solids
but only longitudinal waves can propagate
through fluids.
9. When a mechanical wave passes through an elastic medium,
the displacement of any particle of the medium at a space
point x at time t is given by the,
y (x, t) = f (x โ v t)
If wave is performing S.H.M
โด ๐ ๐, ๐ = ๐จ ๐๐๐ ๐ ๐ ยฑ ๐ ๐
Reflection of Waves
โข When a progressive wave, travelling through a medium, reaches an interface
separating two media, a certain part of the wave energy comes back in the same
medium. The wave changes its direction of travel. This is called reflection of a wave
from the interface.
โข Reflection is the phenomenon in which the sound wave traveling from one medium to
another comes back in the original medium with slightly different intensity and energy.
10. Reflection of a Transverse Wave
โข Newton third law of motion
applied.
โข Crest reflected as trough
โข Trough reflected as crest
โข Wave is seen to be reflected
back as crest.
โข There is no change in phase.
11. Reflection of a Transverse Wave
โข Crest travelling from heavy
string get reflected as crest
from light string.
โข When travel from denser
medium to rarer medium it
cases to reflected as crest as
crest.
12. Reflection of a Longitudinal Wave
โข When longitudinal wave travels
from a rarer medium to a
denser medium:
โข Compression โ Compression
โข Rarefaction โ Rarefaction
โข When longitudinal wave travels
from a denser medium to a
rarer medium:
โข Compression โ Rarefaction
โข Rarefaction โ Compression
13. Superposition of Waves
โโwhen two or more waves, travelling through a medium, pass
through a common point, each wave produces its own
displacement at that point, independent of the presence of the
other wave. The resultant displacement at that point is equal to
the vector sum of the displacements due to the individual wave
at that point.โโ
14. Superposition of Two Wave Pulses of Equal Amplitude and Same Phase
Moving towards Each Other
Superposition of two wave pulses of equal amplitude and opposite
phases moving towards each other.
15. Amplitude of theResultant Wave Produced due to Superpositionof Two Waves
๐ฆ1 = ๐ด1 sin ๐๐ก
๐ฆ2 = ๐ด2 sin (๐๐ก + ๐)
According to principle of superposition;
๐ฆ = ๐ฆ1 + ๐ฆ2
๐ฆ = (๐ด1 + ๐ด2 cos ๐) sin ๐๐ก + ๐ด2 cos ๐๐ก. sin ๐
๐ด1 + ๐ด2 cos ๐ = ๐ด cos ๐ โฆโฆโฆโฆโฆโฆโฆโฆโฆโฆ.(i)
๐ด2 sin ๐ = ๐ด sin ๐ โฆโฆโฆโฆโฆโฆโฆโฆโฆโฆ.(ii)
๐ฆ = ๐ด cos ๐. sin ๐๐ก + ๐ด sin ๐. cos ๐๐ก
๐ฆ = sin(๐๐ก + ๐) โฆโฆโฆโฆโฆ.(iii)
For resultant amplitude,
โด ๐ด = ๐ด1
2
+ 2๐ด1๐ด2 cos ๐ + ๐ด2
2
17. Stationary Waves
Equationof Stationary Wave on a StretchedString
Consider two simple harmonic progressive waves of equal amplitudes (a)
and wavelength (ฮป)
We know that,
๐ ๐
๐
= ๐ and ๐ =
๐ ๐
๐ป
i.e., ๐ = ๐ ๐ ๐
๐ = ๐จ ๐๐๐ ๐๐
18. Stationary Waves
When two identical waves travelling along the same path in
opposite directions interfere with each other, resultant wave is
called stationary wave.
19. Condition for node
Nodes are the points of minimum displacement.
This is possible if the amplitude is minimum (zero), i.e.
๐ = ๐,
๐
๐
, ๐,
๐๐
๐
, ๐๐, โฆ โฆ โฆ โฆ
Commonly, ๐ =
๐๐
๐
, Where ๐ = ๐, ๐, ๐, ๐, โฆ .
Note - The distance between two successive nodes is
๐
๐
.
Condition for antinode
Antinodes are the points of maximum displacement, A = ยฑ๐๐
๐ =
๐
๐
,
๐๐
๐
,
๐๐
๐
,
๐๐
๐
, โฆ โฆ โฆ โฆ
Commonly, ๐ =
๐(๐๐โ๐)
๐
, Where ๐ = ๐, ๐, ๐, โฆ .
Note - The distance between two successive antinodes is
๐
๐
.
20. At the end of tube
โข The distance between two successive antinodes
is
๐
๐
.
โข Nodes and antinodes are formed alternately.
โข Therefore, the distance between a node and an
adjacent antinode is
๐
๐
.
โข Longitudinal wave will behave like transverse
wave.
โข Antinode โ Maximum displacement
โข Node โ Minimum displacement
21. Properties of Stationary Waves
1. Stationary waves are produced due to superposition of two identical waves traveling through a medium along the
same path in opposite directions.
2. If two identical transverse progressive waves superimpose or interfere, the resultant wave is a transverse stationary
wave.
โข When a transverse stationary wave is produced on a string, some points on the string are motionless. The points which
do not move are called nodes.
โข There are some points on the string which oscillate with greatest amplitude (say A). They are called antinodes.
โข Points between the nodes and antinodes vibrate with values of amplitudes between 0 and A.
3. If two identical longitudinal progressive waves superimpose or interfere, the resultant wave is a longitudinal stationary
wave. Stationary sound wave produced in a pipe closed at one end.
โข The points, at which the amplitude of the particles of the medium is minimum (zero), are called nodes.
โข The points, at which the amplitude of the particles of the medium is maximum (say A), are called antinodes.
โข Points between the nodes and antinodes vibrate with values of amplitudes between 0 and A
4. The distance between two consecutive nodes is ฮป/2 and the distance between two consecutive antinodes is ฮป/2.
5. Nodes and antinodes are produced alternately. The distance between a node and an adjacent antinode is ฮป/4.
6. The amplitude of vibration varies periodically in space. All points vibrate with the same frequency.
7. Though all the particles possess energy, there is no propagation of energy. The wave is localized and its velocity is zero.
Therefore, we call it a stationary wave.
8. All the particles between adjacent nodes vibrate in phase. There is no progressive change of phase from one particle to
another particle. All the particles in the same loop are in the same phase of oscillation, which reverses for the adjacent
loop.
22. โข Find the distance between two successive nodes in a stationary
wave on a string vibrating with frequency 64 Hz. The velocity of
progressive wave that resulted in the stationary wave is 48
m/s.
Solution: Given: Speed of wave = v = 48 m/s
Frequency n = 64 Hz
We have v = n ๐
โด ฮป = v/n = 48/64
= 0.75 m
We know that distance between successive nodes =
๐
๐
= 0 75/2
= 0.375 m
23. Comparison of Progressive Waves and Stationary Waves
1. In a progressive wave, the disturbance travels form one region to the other with
definite velocity. In stationary waves, disturbance remains in the region where it is
produced, velocity of the wave is zero.
2. In progressive waves, amplitudes of all particles are same but in stationary waves,
amplitudes of particles are different.
3. In a stationary wave, all the particles cross their mean positions simultaneously
but in a progressive wave, this does not happen.
4. In progressive waves, all the particles are moving while in stationary waves
particles at the position of nodes are always at rest.
5. Energy is transmitted from one region to another in progressive waves but in
stationary waves there is no transfer of energy.
6. All particles between two consecutive nodes are moving in the same direction and
are in phase while those in adjacent loops are moving in opposite directions and
differ in phase by 180ยฐ in stationary waves but in a progressive wave, phases of
adjacent particles are different.
24. Free and Forced Vibrations
โข In free vibration, the body at first is given an initial displacement and
the force is then withdrawn.
โข The body starts vibrating and continues the motion on its own.
โข No external force acts on the body further to keep it in motion.
Forced Vibrations
โข When body is moving or vibrating with
external force.
โข Period of movement (frequency) is
different from its natural frequency.
CONCLUSION
โข Frequency is different from natural frequency.
โข If frequency matched, it is termed as resonance.
25. Harmonics and Overtones
Fundamental
Frequency
The minimum frequency of wave at which a loop of a wave is created.
n โ Fundamental frequency, v = n ๐
1st harmonic
โข Integer multiple of fundamental frequency โ Harmonic
โข Frequency more then fundamental frequency โ Overtones
โข The first frequency higher than the
fundamental frequency is called the first
overtone, the next frequency higher is the
second overtone and so on.
26. End Correction
โข Antinode always forms at open
end of pipe.
โข The distance between the open
end of the pipe and the position of
antinode is called the end
correction.
โข According to Reynold, the end
correction at an end is,
e = 0.3 d
โข where d is the inner diameter of
the pipe.
27. End Correction
For a pipe open at both ends
โข The corrected length of air column
โข L = length of air column in pipe l +
end corrections at both the ends.
โข โด L = l + 2e
For a pipe closed at one end
โข The corrected length of air column
โข L = length of air column in pipe l +
end correction at the open end.
โข โด L = l + e
28. Vibrations of air columnin a pipe closed at one end
โข This is the simplest mode of vibration of air
column closed at one end, known as the
fundamental mode.
โข โด Length of air column
โข L =
๐
๐
and ๐ = ๐ ๐ณ
โข V = n ๐
โข ๐ =
๐ฝ
๐
โข ๐ =
๐ฝ
๐ ๐ณ
โข ๐ =
๐ฝ
๐ (๐+๐)
29. Next modeof vibrations
If ๐1 and ๐1 is the frequency and wavelength of the
mode of vibrations.
โด Length of the air column, ๐ฟ =
3๐1
4
๐1 =
4๐ฟ
3
๐1 =
4(๐+๐)
3
Now, Velocity in 2nd mode, ๐ฃ = ๐1๐1
โด ๐1 =
3๐
4๐ฟ
โด ๐1 = 3๐
30. Next higher modeof vibrations
If ๐2 and ๐2 is the frequency and wavelength of the mode of
vibrations.
โด Length of the air column, ๐ฟ =
5๐2
4
๐2 =
4๐ฟ
5
๐2 =
4(๐+๐)
5
Now, ๐ฃ = ๐2๐2
โด ๐2 =
5๐
4๐ฟ
โด ๐2 = 5๐
For ๐๐กโ overtone,
Frequency, ๐๐ = 2๐ + 1 ๐
31. Vibrations of air columnin a pipe openat both ends
In this condition, antinode is present on both side of the end.
L = l + 2 e
Now, for 1st mode of a oscillation,
If n and ๐ is the frequency and wavelength.
โด Length of the air column, ๐ฟ =
๐
2
โด ๐ = 2 ๐ฟ
Now, ๐ฃ = ๐ ๐
โด ๐ =
๐ฃ
๐
โฆโฆโฆโฆ.. (1st Harmonic)
โด ๐ =
๐ฃ
2๐ฟ
โฆโฆโฆ (i)
โด ๐ =
๐ฃ
2(๐+2๐)
32. Next POSSIBLEmodeof vibrations
If ๐1 and ๐1 is the frequency and wavelength of the
mode of vibrations.
โด Length of the air column, ๐ฟ = ๐1
Now, ๐ฃ = ๐1๐1
โด ๐1 =
๐
๐1
โด ๐1 =
๐
๐ฟ
Now, from equation (i)
โด ๐ =
๐1
2
โด ๐1 = 2 ๐ โฆโฆโฆ(2nd Harmonic / 1st Overtone)
33. FOR Next HIGHER mode of vibrations
If ๐2 and ๐2 is the frequency and wavelength of the mode of vibrations.
โด Length of the air column, ๐ฟ =
3๐2
2
๐2 =
2๐ฟ
3
Now, ๐ฃ = ๐2๐2
โด ๐2 =
3๐
2๐ฟ
From equation (i)
โด ๐2 = 3๐ โฆโฆโฆโฆ(3rd Harmonic / 2nd Overtone)
For ๐๐กโ overtone,
Frequency, ๐๐ = ๐ + 1 ๐
34. It may be noted that,
1. Sound produced by an open pipe contains all
harmonics. Its quality is richer than that produced
by a closed pipe.
2. Fundamental frequency of vibration of air column
in an open pipe is double that of the fundamental
frequency of vibration in a closed pipe of the same
length.
35. Practical Determinationof End Correction
Two pipes of same diameter but different lengths ๐๐ and ๐๐
๐ =
๐ฃ
4๐ฟ
โข For a pipe open at both ends:
๐ = ๐๐๐๐ณ๐ = ๐๐๐๐ณ๐
๐๐ ๐๐ + ๐๐ = ๐๐ ๐๐ + ๐๐
2e(๐๐ โ ๐๐) = ๐๐๐๐ โ ๐๐๐๐
๐ =
๐๐๐๐โ๐๐๐๐
๐(๐๐โ๐๐)
โข For a pipe closed at one end:
๐ = ๐๐๐๐ณ๐ = ๐๐๐๐ณ๐
๐๐ ๐๐ + ๐ = ๐๐ ๐๐ + ๐
e(๐๐ โ ๐๐) = ๐๐๐๐ โ ๐๐๐๐
๐ =
๐๐๐๐โ๐๐๐๐
(๐๐โ๐๐)
36. Vibrations Producedon a String
Consider, l โ Length of string, m โ linear density, T โ Tension
For fundamental frequency, ๐ =
๐
๐
โด ๐ = ๐ ๐
Now, we know, v = n ๐
โด ๐ =
๐
๐
=
๐
๐๐
๐ =
๐
๐๐
๐ป
๐
โฆโฆโฆโฆ(i)
โข Total length of loop, ๐ = ๐๐
Now, we know, v = ๐๐๐๐
Now, ๐๐ =
๐
๐
๐๐ =
๐
๐
๐ป
๐
From equation (i), ๐๐ = ๐๐ โฆโฆ.(2nd Harmonic / 1st Overtone)
37. โข Total length of loop, ๐ =
๐๐๐
๐
Now, we know, v = ๐๐๐๐
Now, ๐๐ =
๐
๐๐
๐๐ =
๐
๐๐
๐ป
๐
From equation (i), ๐๐ = ๐๐ โฆโฆ.(3rd Harmonic / 2nd Overtone)
โข For Higher mode of vibration
For ๐๐๐
overtone on string,
๐๐ = ๐ + ๐ ๐
38. Laws of a Vibrating String
The fundamental frequency of a vibrating string under tension is given as
๐ =
๐
๐๐
๐ป
๐
1) Law of length: ๐ ๐ถ
๐
๐
2) Law of tension: ๐ ๐ถ ๐ป
3) Law of linear density: ๐ ๐ถ
๐
๐
Linear density = mass per unit length = volume per unit length ร density
=
๐ ๐๐๐
๐
๐
๐จ๐ ๐ ๐ถ
๐
๐
, , if T and l are constant, we get ๐ ๐ถ
๐
๐ ๐๐๐
Thus the fundamental frequency of vibrations of a stretched string is
inversely proportional to (i) the radius of string and (ii) the square root of
the density of the material of vibrating string.
39. Sonometer
โข A sonometer consists of a hollow
rectangular wooden box called
the sound box.
โข The sound box is used to make a
larger mass of air vibrate so that
the sound produced by the
vibrating string (metal wire in this
case) gets amplified.
โข The same principle is applied in
stringed instruments such as the
violin, guitar, tanpura etc.
40.
41. 1) Verification of first law of a vibrating string:
By measuring length of wire
๐๐๐๐ = ๐๐๐๐ = ๐๐๐๐ =โฆโฆโฆโฆโฆ..(constant)
Finally, n l = constant
๐. ๐. ๐ ๐ถ
๐
๐
โฆโฆโฆโฆ..where T and m are constant
2) Verification of second law of a vibrating string:
The vibrating length (l) of the given wire of mass per unit length (m) is kept
constant. If tensions ๐ป๐, ๐ป๐, ๐ป๐, โฆ.. correspond to frequencies ๐๐, ๐๐, ๐๐ ,.....etc.
So,
๐๐
๐ป๐
,
๐๐
๐ป๐
,
๐๐
๐ป๐
, โฆ โฆ . . = ๐๐๐๐๐๐๐๐
โด
๐
๐ป
= ๐๐๐๐๐๐๐๐
โด ๐ ๐ถ ๐ป
3) Verification of third law of a vibrating string:
two wires having different masses per unit lengths ๐๐ and ๐๐ (linear densities)
โด ๐๐ ๐๐ = ๐๐ ๐๐
โด ๐ ๐ = ๐๐๐๐๐๐๐๐
โด ๐ โ
๐
๐
42. Beats
โข This is an interesting phenomenon based on the principle of
superposition of waves.
โข When there is superposition of two sound waves, having same
amplitude but slightly different frequencies, travelling in the same
direction, the intensity of sound varies periodically with time. This
phenomenon is known as production of beats.
โข Waxing
The occurrences of maximum intensity are called waxing
โข Waning
The occurrences of minimum intensity are called waning
NOTE:
โข One waxing and successive waning together constitute one beat.
โข The number of beats heard per second is called beat frequency.
43. Analytical methodto determine beat frequency
Two sound waves, having same amplitude and slightly different
frequencies ๐๐ and ๐๐.
Let us assume for simplicity that the listener is at x = 0.
According to the principle of superposition of waves, ๐ = ๐๐ + ๐๐
๐ = ๐จ ๐ฌ๐ข๐ง(๐ ๐ ๐ ๐)
This is the equation of a progressive wave having frequency n and
amplitude A.
The intensity of sound is proportional to the square of the
amplitude. Hence the resultant intensity will be maximum when the
amplitude is maximum.
โข For maximum amplitude (waxing)
โข For minimum amplitude, A = 0
44. โข For maximum amplitude (waxing), A = ยฑ๐๐
๐ = ๐,
๐
๐๐โ๐๐
,
๐
๐๐โ๐๐
,
๐
๐๐โ๐๐
, โฆ โฆ โฆ โฆ . .
Thus, the time interval between two successive maxima of sound is always
๐
๐๐โ๐๐
.
Period of beats, T =
๐
๐๐โ๐๐
The number of waxing heard per second is the reciprocal of period of waxing.
โด frequency of beats, n = ๐๐ โ ๐๐
โข For minimum amplitude (waning), A = ๐
๐ =
๐
๐(๐๐โ๐๐)
,
๐
๐(๐๐โ๐๐)
,
๐
๐(๐๐โ๐๐)
, โฆ โฆ โฆ โฆ . .
Thus, the time interval between two successive minima of sound is
๐
๐๐โ๐๐
.
Period of beats, T =
๐
๐๐โ๐๐
45. Applications of beats
โข The phenomenon of beats is
used for matching the
frequencies of different
musical instruments by artists.
โข The speed of an airplane can
be determined by using
Doppler RADAR.
โข Unknown frequency of a
sound note can be
determined by using the
phenomenon of beats.
https://thefactfactor.com/facts/pure_science/physics/formation-of-beats/6682/
46. Characteristics of Sound
1. Loudness:
โข Loudness is the human perception to intensity of sound.
โข 1 decibel = 1/10 bel
โข Loudness is different at different frequencies, even for the same intensity.
โข For measuring loudness the unit phon is used. Phon is a measure of loudness.
2. Pitch:
โข It is a sensation of sound which helps the listener to distinguish between a high
frequency and a low frequency note.
โข Pitch is the human perception to frequency- higher frequency denotes higher pitch.
โข The pitch of a female voice is higher than that of a male voice.
3. Quality or timbre:
โข Quality of sound is that characteristic which enables us to distinguish between two
sounds of same pitch and loudness.
โข A sequence of frequencies which have a specific relationship with each other is
called a musical scale.
47. MUSICAL INSTRUMENT
Three main types
Stringed
Instruments
Consist of stretched string
Sound produced by plucking of string
Plucked
String Type
Tanpura, sitar, guitar,
etc.
Bowed
String Type
Violin, sarangi.
Struck String
Type
Santoor, piano.
Wind
Instrument
Consist of air column
Freewind
Type
Mouth
organ,
harmonium
etc.
Edge Type
Flute.
Reedpipes
Saxophone,
bassoon, etc.
Percussion
Instrument
dhol, sambal, etc.