SUBJECT: PHYSICS - Chapter 6 : Superposition of waves (CLASS XII - MAHARASHTRA STATE BOARD)

Pooja M
Pooja MTeacher en Pooja M
SUBJECT : PHYSICS
Chapter 6 : Superposition of waves
CLASSXII
MAHARASHTRA STATE BOARD
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?
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 (𝝀)
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.
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.
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.
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.
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.
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 – v t)
If wave is performing S.H.M
∴ 𝒚 𝒙, 𝒕 = 𝑨 𝒔𝒊𝒏 𝒌 𝒙 ± 𝒘 𝒕
Reflection of Waves
• 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.
• 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.
Reflection of a Transverse Wave
• Newton third law of motion
applied.
• Crest reflected as trough
• Trough reflected as crest
• Wave is seen to be reflected
back as crest.
• There is no change in phase.
Reflection of a Transverse Wave
• Crest travelling from heavy
string get reflected as crest
from light string.
• When travel from denser
medium to rarer medium it
cases to reflected as crest as
crest.
Reflection of a Longitudinal Wave
• When longitudinal wave travels
from a rarer medium to a
denser medium:
• Compression → Compression
• Rarefaction → Rarefaction
• When longitudinal wave travels
from a denser medium to a
rarer medium:
• Compression → Rarefaction
• Rarefaction → Compression
Superposition of Waves
‘’when 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.’’
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.
Amplitude of theResultant Wave Produced due to Superpositionof Two Waves
𝑦1 = 𝐴1 sin 𝜔𝑡
𝑦2 = 𝐴2 sin (𝜔𝑡 + 𝜙)
According to principle of superposition;
𝑦 = 𝑦1 + 𝑦2
𝑦 = (𝐴1 + 𝐴2 cos 𝜙) sin 𝜔𝑡 + 𝐴2 cos 𝜔𝑡. sin 𝜙
𝐴1 + 𝐴2 cos 𝜙 = 𝐴 cos 𝜃 ………………………….(i)
𝐴2 sin 𝜙 = 𝐴 sin 𝜃 ………………………….(ii)
𝑦 = 𝐴 cos 𝜃. sin 𝜔𝑡 + 𝐴 sin 𝜃. cos 𝜔𝑡
𝑦 = sin(𝜔𝑡 + 𝜙) …………….(iii)
For resultant amplitude,
∴ 𝐴 = 𝐴1
2
+ 2𝐴1𝐴2 cos 𝜙 + 𝐴2
2
Special cases:
(i)If 𝜙 = 0
𝐴 = 𝐴1
2
+ 2𝐴1𝐴2 cos 𝜙 + 𝐴2
2
𝐴 = (𝐴1 + 𝐴2)2
(i)If 𝜙 = 𝜋
𝐴 = 𝐴1
2
+ 2𝐴1𝐴2 cos 𝜙 + 𝐴2
2
𝐴 = (𝐴1 − 𝐴2)2
Intensities of the wave
∴ 𝐼𝑚𝑎𝑥 𝛼 (𝐴𝑚𝑎𝑥)2
∴ 𝐼𝑚𝑖𝑛 𝛼 (𝐴𝑚𝑖𝑛)2
Stationary Waves
Equationof Stationary Wave on a StretchedString
Consider two simple harmonic progressive waves of equal amplitudes (a)
and wavelength (λ)
We know that,
𝟐 𝝅
𝝀
= 𝒌 and 𝝎 =
𝟐 𝝅
𝑻
i.e., 𝝎 = 𝟐 𝝅 𝒏
𝒚 = 𝑨 𝒔𝒊𝒏 𝝎𝒕
Stationary Waves
When two identical waves travelling along the same path in
opposite directions interfere with each other, resultant wave is
called stationary wave.
Condition for node
Nodes are the points of minimum displacement.
This is possible if the amplitude is minimum (zero), i.e.
𝒙 =
𝝀
𝟒
,
𝟑𝝀
𝟒
,
𝟓𝝀
𝟒
, … … … …
Commonly, 𝒙 = (𝟐𝒑 − 𝟏)
𝝀
𝟒
, Where 𝒑 = 𝟏, 𝟐, 𝟑, … .
Note - The distance between two successive nodes is
𝝀
𝟐
.
Condition for antinode
Antinodes are the points of maximum displacement, A = ±𝟐𝒂
𝒙 = 𝟎,
𝝀
𝟐
,
𝟑𝝀
𝟐
, 𝟐𝝀, … … … …
Commonly, 𝒙 =
𝝀𝒑
𝟒
, Where 𝒑 = 𝟎, 𝟏, 𝟐, 𝟑, … .
Note - The distance between two successive antinodes is
𝝀
𝟐
.
At the end of tube
• The distance between two successive antinodes
is
𝝀
𝟐
.
• Nodes and antinodes are formed alternately.
• Therefore, the distance between a node and an
adjacent antinode is
𝝀
𝟒
.
• Longitudinal wave will behave like transverse
wave.
• Antinode – Maximum displacement
• Node – Minimum displacement
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.
• 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.
• There are some points on the string which oscillate with greatest amplitude (say A). They are called antinodes.
• 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.
• The points, at which the amplitude of the particles of the medium is minimum (zero), are called nodes.
• The points, at which the amplitude of the particles of the medium is maximum (say A), are called antinodes.
• Points between the nodes and antinodes vibrate with values of amplitudes between 0 and A
4. The distance between two consecutive nodes is λ/2 and the distance between two consecutive antinodes is λ/2.
5. Nodes and antinodes are produced alternately. The distance between a node and an adjacent antinode is λ/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.
• 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 𝝀
∴ λ = v/n = 48/64
= 0.75 m
We know that distance between successive nodes =
𝝀
𝟐
= 0 75/2
= 0.375 m
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 180° in stationary waves but in a progressive wave, phases of
adjacent particles are different.
Free and Forced Vibrations
• In free vibration, the body at first is given an initial displacement and
the force is then withdrawn.
• The body starts vibrating and continues the motion on its own.
• No external force acts on the body further to keep it in motion.
Forced Vibrations
• When body is moving or vibrating with
external force.
• Period of movement (frequency) is
different from its natural frequency.
CONCLUSION
• Frequency is different from natural frequency.
• If frequency matched, it is termed as resonance.
Harmonics and Overtones
Fundamental
Frequency
The minimum frequency of wave at which a loop of a wave is created.
n → Fundamental frequency, v = n 𝝀
1st harmonic
• Integer multiple of fundamental frequency → Harmonic
• Frequency more then fundamental frequency → Overtones
• The first frequency higher than the
fundamental frequency is called the first
overtone, the next frequency higher is the
second overtone and so on.
End Correction
• Antinode always forms at open
end of pipe.
• The distance between the open
end of the pipe and the position of
antinode is called the end
correction.
• According to Reynold, the end
correction at an end is,
e = 0.3 d
• where d is the inner diameter of
the pipe.
End Correction
For a pipe open at both ends
• The corrected length of air column
• L = length of air column in pipe l +
end corrections at both the ends.
• ∴ L = l + 2e
For a pipe closed at one end
• The corrected length of air column
• L = length of air column in pipe l +
end correction at the open end.
• ∴ L = l + e
Vibrations of air columnin a pipe closed at one end
• This is the simplest mode of vibration of air
column closed at one end, known as the
fundamental mode.
• ∴ Length of air column
• L =
𝝀
𝟒
and 𝝀 = 𝟒 𝑳
• V = n 𝝀
• 𝒏 =
𝑽
𝝀
• 𝒏 =
𝑽
𝟒 𝑳
• 𝒏 =
𝑽
𝟒 (𝒍+𝒆)
Next modeof vibrations
If 𝑛1 and 𝜆1 is the frequency and wavelength of the
mode of vibrations.
∴ Length of the air column, 𝐿 =
3𝜆1
4
𝜆1 =
4𝐿
3
𝜆1 =
4(𝑙+𝑒)
3
Now, Velocity in 2nd mode, 𝑣 = 𝑛1𝜆1
∴ 𝑛1 =
3𝑉
4𝐿
∴ 𝑛1 = 3𝑛
Next higher modeof vibrations
If 𝑛2 and 𝜆2 is the frequency and wavelength of the mode of
vibrations.
∴ Length of the air column, 𝐿 =
5𝜆2
4
𝜆2 =
4𝐿
5
𝜆1 =
4(𝑙+𝑒)
5
Now, 𝑣 = 𝑛2𝜆2
∴ 𝑛2 =
5𝑉
4𝐿
∴ 𝑛2 = 5𝑛
For 𝑝𝑡ℎ overtone,
Frequency, 𝑛𝑝 = 2𝑝 + 1 𝑛
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 𝜆 is the frequency and wavelength.
∴ Length of the air column, 𝐿 =
𝜆
2
∴ 𝜆 = 2 𝐿
Now, 𝑣 = 𝑛 𝜆
∴ 𝑛 =
𝑣
𝜆
………….. (1st Harmonic)
∴ 𝑛 =
𝑣
2𝐿
……… (i)
∴ 𝑛 =
𝑣
2(𝑙+2𝑒)
Next POSSIBLEmodeof vibrations
If 𝑛1 and 𝜆1 is the frequency and wavelength of the
mode of vibrations.
∴ Length of the air column, 𝐿 = 𝜆1
Now, 𝑣 = 𝑛1𝜆1
∴ 𝑛1 =
𝑉
𝜆1
∴ 𝑛1 =
𝑉
𝐿
Now, from equation (i)
∴ 𝑛 =
𝑛1
2
∴ 𝑛1 = 𝑛 ………(2nd Harmonic / 1st Overtone)
FOR Next HIGHER mode of vibrations
If 𝑛2 and 𝜆2 is the frequency and wavelength of the mode of vibrations.
∴ Length of the air column, 𝐿 =
3𝜆2
2
𝜆2 =
2𝐿
3
Now, 𝑣 = 𝑛2𝜆2
∴ 𝑛2 =
3𝑉
2𝐿
From equation (i)
∴ 𝑛2 = 3𝑛 …………(3rd Harmonic / 2nd Overtone)
For 𝑝𝑡ℎ overtone,
Frequency, 𝑛𝑝 = 𝑝 + 1 𝑛
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.
Practical Determinationof End Connection
Two pipes of same diameter but different lengths 𝒍𝟏 and 𝒍𝟐
𝑛 =
𝑣
4𝐿
• For a pipe open at both ends:
𝒗 = 𝟐𝒏𝟏𝑳𝟏 = 𝟐𝒏𝟐𝑳𝟐
𝒏𝟏 𝒍𝟏 + 𝟐𝒆 = 𝒏𝟐 𝒍𝟐 + 𝟐𝒆
2e(𝒏𝟏 − 𝒏𝟐) = 𝒏𝟐𝒍𝟐 − 𝒏𝟏𝒍𝟏
𝒆 =
𝒏𝟐𝒍𝟐−𝒏𝟏𝒍𝟏
𝟐(𝒏𝟏−𝒏𝟐)
• For a pipe closed at one end:
𝒗 = 𝟒𝒏𝟏𝑳𝟏 = 𝟒𝒏𝟐𝑳𝟐
𝒏𝟏 𝒍𝟏 + 𝒆 = 𝒏𝟐 𝒍𝟐 + 𝒆
e(𝒏𝟐 − 𝒏𝟏) = 𝒏𝟏𝒍𝟏 − 𝒏𝟐𝒍𝟐
𝒆 =
𝒏𝟏𝒍𝟏−𝒏𝟐𝒍𝟐
(𝒏𝟐−𝒏𝟏)
Vibrations Producedin a String
Consider, l → Length of string, m → linear density, T → Tension
For fundamental frequency, 𝒍 =
𝝀
𝟐
∴ 𝝀 = 𝟐 𝒍
Now, we know, v = n 𝝀
∴ 𝒏 =
𝒗
𝝀
=
𝒗
𝟐𝒍
𝒏 =
𝟏
𝟐𝒍
𝑻
𝒎
• Total length of loop, 𝒍 = 𝝀𝟏
Now, we know, v = 𝒏𝟏𝝀𝟏
Now, 𝒏𝟏 =
𝒗
𝒍
𝒏𝟏 =
𝟏
𝒍
𝑻
𝒎
From equation (i), 𝒏𝟏 = 𝟐𝒏 …….(2nd Harmonic / 1st Overtone)
• Total length of loop, 𝒍 =
𝟑𝝀𝟐
𝟐
Now, we know, v = 𝒏𝟐𝝀𝟐
Now, 𝒏𝟐 =
𝒗
𝝀𝟐
𝒏𝟐 =
𝟑
𝟐𝒍
𝑻
𝒎
From equation (i), 𝒏𝟐 = 𝟑𝒏 …….(3rd Harmonic / 2nd Overtone)
• For Higher mode of vibration
For 𝒑𝒕𝒉
overtone on string,
𝒏𝒑 = 𝒑 + 𝟏 𝒏
Where, p → Fundamental Frequency
Laws of a Vibrating String
The fundamental frequency of a vibrating string under tension is given as
𝒏 =
𝟏
𝟐𝒍
𝑻
𝒎
1) Law of length: 𝒏 𝜶
𝟏
𝑻
2) Law of tension: 𝒏 𝜶 𝑻
3) Law of linear density: 𝒏 𝜶
𝟏
𝒎
Linear density = mass per unit length = volume per unit length × density
=
𝝅𝒓𝟐𝒍
𝒍
𝝆
𝑨𝒔 𝒏 𝜶
𝟏
𝒎
, , if T and l are constant, we get 𝒏 𝜶
𝟏
𝝅𝒓𝟐𝝆
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.
Sonometer
• A sonometer consists of a hollow
rectangular wooden box called
the sound box.
• 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.
• The same principle is applied in
stringed instruments such as the
violin, guitar, tanpura etc.
 SUBJECT: PHYSICS - Chapter 6 : Superposition of waves  (CLASS XII - MAHARASHTRA STATE BOARD)
1) Verification of first law of a vibrating string:
By measuring length of wire
𝒏𝟏𝒍𝟏 = 𝒏𝟐𝒍𝟐 = 𝒏𝟑𝒍𝟑 =……………..(constant)
Finally, n l = constant
𝒊. 𝒆. 𝒏 𝜶
𝟏
𝒍
…………..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 𝑻𝟏, 𝑻𝟐, 𝑻𝟑, ….. correspond to frequencies 𝒏𝟏, 𝒏𝟐, 𝒏𝟑 ,.....etc.
So,
𝒏𝟏
𝑻𝟏
,
𝒏𝟐
𝑻𝟐
,
𝒏𝟑
𝑻𝟑
, … … . . = 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕
∴
𝒏
𝑻
= 𝒄𝒐𝒏𝒔𝒕𝒔𝒏𝒕
∴ 𝒏 𝜶 𝑻
3) Verification of third law of a vibrating string:
two wires having different masses per unit lengths 𝒎𝟏 and 𝒎𝟐 (linear densities)
∴ 𝒍𝟏 𝒎𝟏 = 𝒍𝟐 𝒎𝟐
∴ 𝒍 𝒎 = 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕
∴ 𝒏 ∝
𝟏
𝒎
Beats
• This is an interesting phenomenon based on the principle of
superposition of waves.
• 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.
• Waxing
The occurrences of maximum intensity are called waxing
• Waning
The occurrences of minimum intensity are called waning
NOTE:
• One waxing and successive waning together constitute one beat.
• The number of beats heard per second is called beat frequency.
Analytical methodto determine beat frequency
Two sound waves, having same amplitude and slightly different
frequencies 𝒏𝟏 and 𝒏𝟐.
Let us assume for simplicity that the listener is at x = 0.
According to the principle of superposition of waves, 𝒚 = 𝒚𝟏 + 𝒚𝟐
𝒚 = 𝑨 𝐬𝐢𝐧(𝟐 𝝅 𝒏 𝒕)
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.
• For maximum amplitude (waxing)
• For minimum amplitude, A = 0
• For maximum amplitude (waxing), A = ±𝑨
𝒕 = 𝟎,
𝟏
𝒏𝟏−𝒏𝟐
,
𝟐
𝒏𝟏−𝒏𝟐
,
𝟑
𝒏𝟏−𝒏𝟐
, … … … … . .
Thus, the time interval between two successive maxima of sound is always
𝟐
𝒏𝟏−𝒏𝟐
.
Period of beats, T =
𝟐
𝒏𝟏−𝒏𝟐
The number of waxing heard per second is the reciprocal of period of waxing.
∴ frequency of beats, N = 𝒏𝟏 − 𝒏𝟐
• For minimum amplitude (waning), A = 𝟎
𝒕 =
𝟏
𝟐(𝒏𝟏−𝒏𝟐)
,
𝟑
𝟐(𝒏𝟏−𝒏𝟐)
,
𝟓
𝟐(𝒏𝟏−𝒏𝟐)
, … … … … . .
Thus, the time interval between two successive minima of sound is
𝟏
𝒏𝟏−𝒏𝟐
.
Period of beats, T =
𝟐
𝒏𝟏−𝒏𝟐
Applications of beats
1) The phenomenon of beats is
used for matching the
frequencies of different musical
instruments by artists.
2) The speed of an airplane can
be determined by using Doppler
RADAR.
3) Unknown frequency of a
sound note can be determined
by using the phenomenon of
beats.
Characteristics of Sound
1. Loudness:
• Loudness is the human perception to intensity of sound.
• 1 decibel = 1/10 bel
• Loudness is different at different frequencies, even for the same intensity.
• For measuring loudness the unit phon is used. Phon is a measure of loudness.
2. Pitch:
• It is a sensation of sound which helps the listener to distinguish between a high
frequency and a low frequency note.
• Pitch is the human perception to frequency- higher frequency denotes higher pitch.
• The pitch of a female voice is higher than that of a male voice.
3. Quality or timbre:
• Quality of sound is that characteristic which enables us to distinguish between two
sounds of same pitch and loudness.
• A sequence of frequencies which have a specific relationship with each other is
called a musical scale.
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.
Thank You
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SUBJECT: PHYSICS - Chapter 6 : Superposition of waves (CLASS XII - MAHARASHTRA STATE BOARD)

  • 1. 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 (𝝀)
  • 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 – v t) If wave is performing S.H.M ∴ 𝒚 𝒙, 𝒕 = 𝑨 𝒔𝒊𝒏 𝒌 𝒙 ± 𝒘 𝒕 Reflection of Waves • 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. • 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 • Newton third law of motion applied. • Crest reflected as trough • Trough reflected as crest • Wave is seen to be reflected back as crest. • There is no change in phase.
  • 11. Reflection of a Transverse Wave • Crest travelling from heavy string get reflected as crest from light string. • When travel from denser medium to rarer medium it cases to reflected as crest as crest.
  • 12. Reflection of a Longitudinal Wave • When longitudinal wave travels from a rarer medium to a denser medium: • Compression → Compression • Rarefaction → Rarefaction • When longitudinal wave travels from a denser medium to a rarer medium: • Compression → Rarefaction • Rarefaction → Compression
  • 13. Superposition of Waves ‘’when 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.’’
  • 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 𝑦1 = 𝐴1 sin 𝜔𝑡 𝑦2 = 𝐴2 sin (𝜔𝑡 + 𝜙) According to principle of superposition; 𝑦 = 𝑦1 + 𝑦2 𝑦 = (𝐴1 + 𝐴2 cos 𝜙) sin 𝜔𝑡 + 𝐴2 cos 𝜔𝑡. sin 𝜙 𝐴1 + 𝐴2 cos 𝜙 = 𝐴 cos 𝜃 ………………………….(i) 𝐴2 sin 𝜙 = 𝐴 sin 𝜃 ………………………….(ii) 𝑦 = 𝐴 cos 𝜃. sin 𝜔𝑡 + 𝐴 sin 𝜃. cos 𝜔𝑡 𝑦 = sin(𝜔𝑡 + 𝜙) …………….(iii) For resultant amplitude, ∴ 𝐴 = 𝐴1 2 + 2𝐴1𝐴2 cos 𝜙 + 𝐴2 2
  • 16. Special cases: (i)If 𝜙 = 0 𝐴 = 𝐴1 2 + 2𝐴1𝐴2 cos 𝜙 + 𝐴2 2 𝐴 = (𝐴1 + 𝐴2)2 (i)If 𝜙 = 𝜋 𝐴 = 𝐴1 2 + 2𝐴1𝐴2 cos 𝜙 + 𝐴2 2 𝐴 = (𝐴1 − 𝐴2)2 Intensities of the wave ∴ 𝐼𝑚𝑎𝑥 𝛼 (𝐴𝑚𝑎𝑥)2 ∴ 𝐼𝑚𝑖𝑛 𝛼 (𝐴𝑚𝑖𝑛)2
  • 17. Stationary Waves Equationof Stationary Wave on a StretchedString Consider two simple harmonic progressive waves of equal amplitudes (a) and wavelength (λ) We know that, 𝟐 𝝅 𝝀 = 𝒌 and 𝝎 = 𝟐 𝝅 𝑻 i.e., 𝝎 = 𝟐 𝝅 𝒏 𝒚 = 𝑨 𝒔𝒊𝒏 𝝎𝒕
  • 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. 𝒙 = 𝝀 𝟒 , 𝟑𝝀 𝟒 , 𝟓𝝀 𝟒 , … … … … Commonly, 𝒙 = (𝟐𝒑 − 𝟏) 𝝀 𝟒 , Where 𝒑 = 𝟏, 𝟐, 𝟑, … . Note - The distance between two successive nodes is 𝝀 𝟐 . Condition for antinode Antinodes are the points of maximum displacement, A = ±𝟐𝒂 𝒙 = 𝟎, 𝝀 𝟐 , 𝟑𝝀 𝟐 , 𝟐𝝀, … … … … Commonly, 𝒙 = 𝝀𝒑 𝟒 , Where 𝒑 = 𝟎, 𝟏, 𝟐, 𝟑, … . Note - The distance between two successive antinodes is 𝝀 𝟐 .
  • 20. At the end of tube • The distance between two successive antinodes is 𝝀 𝟐 . • Nodes and antinodes are formed alternately. • Therefore, the distance between a node and an adjacent antinode is 𝝀 𝟒 . • Longitudinal wave will behave like transverse wave. • Antinode – Maximum displacement • Node – 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. • 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. • There are some points on the string which oscillate with greatest amplitude (say A). They are called antinodes. • 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. • The points, at which the amplitude of the particles of the medium is minimum (zero), are called nodes. • The points, at which the amplitude of the particles of the medium is maximum (say A), are called antinodes. • Points between the nodes and antinodes vibrate with values of amplitudes between 0 and A 4. The distance between two consecutive nodes is λ/2 and the distance between two consecutive antinodes is λ/2. 5. Nodes and antinodes are produced alternately. The distance between a node and an adjacent antinode is λ/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. • 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 𝝀 ∴ λ = v/n = 48/64 = 0.75 m We know that distance between successive nodes = 𝝀 𝟐 = 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 180° in stationary waves but in a progressive wave, phases of adjacent particles are different.
  • 24. Free and Forced Vibrations • In free vibration, the body at first is given an initial displacement and the force is then withdrawn. • The body starts vibrating and continues the motion on its own. • No external force acts on the body further to keep it in motion. Forced Vibrations • When body is moving or vibrating with external force. • Period of movement (frequency) is different from its natural frequency. CONCLUSION • Frequency is different from natural frequency. • 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 → Fundamental frequency, v = n 𝝀 1st harmonic • Integer multiple of fundamental frequency → Harmonic • Frequency more then fundamental frequency → Overtones • 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 • Antinode always forms at open end of pipe. • The distance between the open end of the pipe and the position of antinode is called the end correction. • According to Reynold, the end correction at an end is, e = 0.3 d • where d is the inner diameter of the pipe.
  • 27. End Correction For a pipe open at both ends • The corrected length of air column • L = length of air column in pipe l + end corrections at both the ends. • ∴ L = l + 2e For a pipe closed at one end • The corrected length of air column • L = length of air column in pipe l + end correction at the open end. • ∴ L = l + e
  • 28. Vibrations of air columnin a pipe closed at one end • This is the simplest mode of vibration of air column closed at one end, known as the fundamental mode. • ∴ Length of air column • L = 𝝀 𝟒 and 𝝀 = 𝟒 𝑳 • V = n 𝝀 • 𝒏 = 𝑽 𝝀 • 𝒏 = 𝑽 𝟒 𝑳 • 𝒏 = 𝑽 𝟒 (𝒍+𝒆)
  • 29. Next modeof vibrations If 𝑛1 and 𝜆1 is the frequency and wavelength of the mode of vibrations. ∴ Length of the air column, 𝐿 = 3𝜆1 4 𝜆1 = 4𝐿 3 𝜆1 = 4(𝑙+𝑒) 3 Now, Velocity in 2nd mode, 𝑣 = 𝑛1𝜆1 ∴ 𝑛1 = 3𝑉 4𝐿 ∴ 𝑛1 = 3𝑛
  • 30. Next higher modeof vibrations If 𝑛2 and 𝜆2 is the frequency and wavelength of the mode of vibrations. ∴ Length of the air column, 𝐿 = 5𝜆2 4 𝜆2 = 4𝐿 5 𝜆1 = 4(𝑙+𝑒) 5 Now, 𝑣 = 𝑛2𝜆2 ∴ 𝑛2 = 5𝑉 4𝐿 ∴ 𝑛2 = 5𝑛 For 𝑝𝑡ℎ overtone, Frequency, 𝑛𝑝 = 2𝑝 + 1 𝑛
  • 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 𝜆 is the frequency and wavelength. ∴ Length of the air column, 𝐿 = 𝜆 2 ∴ 𝜆 = 2 𝐿 Now, 𝑣 = 𝑛 𝜆 ∴ 𝑛 = 𝑣 𝜆 ………….. (1st Harmonic) ∴ 𝑛 = 𝑣 2𝐿 ……… (i) ∴ 𝑛 = 𝑣 2(𝑙+2𝑒)
  • 32. Next POSSIBLEmodeof vibrations If 𝑛1 and 𝜆1 is the frequency and wavelength of the mode of vibrations. ∴ Length of the air column, 𝐿 = 𝜆1 Now, 𝑣 = 𝑛1𝜆1 ∴ 𝑛1 = 𝑉 𝜆1 ∴ 𝑛1 = 𝑉 𝐿 Now, from equation (i) ∴ 𝑛 = 𝑛1 2 ∴ 𝑛1 = 𝑛 ………(2nd Harmonic / 1st Overtone)
  • 33. FOR Next HIGHER mode of vibrations If 𝑛2 and 𝜆2 is the frequency and wavelength of the mode of vibrations. ∴ Length of the air column, 𝐿 = 3𝜆2 2 𝜆2 = 2𝐿 3 Now, 𝑣 = 𝑛2𝜆2 ∴ 𝑛2 = 3𝑉 2𝐿 From equation (i) ∴ 𝑛2 = 3𝑛 …………(3rd Harmonic / 2nd Overtone) For 𝑝𝑡ℎ overtone, Frequency, 𝑛𝑝 = 𝑝 + 1 𝑛
  • 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 Connection Two pipes of same diameter but different lengths 𝒍𝟏 and 𝒍𝟐 𝑛 = 𝑣 4𝐿 • For a pipe open at both ends: 𝒗 = 𝟐𝒏𝟏𝑳𝟏 = 𝟐𝒏𝟐𝑳𝟐 𝒏𝟏 𝒍𝟏 + 𝟐𝒆 = 𝒏𝟐 𝒍𝟐 + 𝟐𝒆 2e(𝒏𝟏 − 𝒏𝟐) = 𝒏𝟐𝒍𝟐 − 𝒏𝟏𝒍𝟏 𝒆 = 𝒏𝟐𝒍𝟐−𝒏𝟏𝒍𝟏 𝟐(𝒏𝟏−𝒏𝟐) • For a pipe closed at one end: 𝒗 = 𝟒𝒏𝟏𝑳𝟏 = 𝟒𝒏𝟐𝑳𝟐 𝒏𝟏 𝒍𝟏 + 𝒆 = 𝒏𝟐 𝒍𝟐 + 𝒆 e(𝒏𝟐 − 𝒏𝟏) = 𝒏𝟏𝒍𝟏 − 𝒏𝟐𝒍𝟐 𝒆 = 𝒏𝟏𝒍𝟏−𝒏𝟐𝒍𝟐 (𝒏𝟐−𝒏𝟏)
  • 36. Vibrations Producedin a String Consider, l → Length of string, m → linear density, T → Tension For fundamental frequency, 𝒍 = 𝝀 𝟐 ∴ 𝝀 = 𝟐 𝒍 Now, we know, v = n 𝝀 ∴ 𝒏 = 𝒗 𝝀 = 𝒗 𝟐𝒍 𝒏 = 𝟏 𝟐𝒍 𝑻 𝒎 • Total length of loop, 𝒍 = 𝝀𝟏 Now, we know, v = 𝒏𝟏𝝀𝟏 Now, 𝒏𝟏 = 𝒗 𝒍 𝒏𝟏 = 𝟏 𝒍 𝑻 𝒎 From equation (i), 𝒏𝟏 = 𝟐𝒏 …….(2nd Harmonic / 1st Overtone)
  • 37. • Total length of loop, 𝒍 = 𝟑𝝀𝟐 𝟐 Now, we know, v = 𝒏𝟐𝝀𝟐 Now, 𝒏𝟐 = 𝒗 𝝀𝟐 𝒏𝟐 = 𝟑 𝟐𝒍 𝑻 𝒎 From equation (i), 𝒏𝟐 = 𝟑𝒏 …….(3rd Harmonic / 2nd Overtone) • For Higher mode of vibration For 𝒑𝒕𝒉 overtone on string, 𝒏𝒑 = 𝒑 + 𝟏 𝒏 Where, p → Fundamental Frequency
  • 38. Laws of a Vibrating String The fundamental frequency of a vibrating string under tension is given as 𝒏 = 𝟏 𝟐𝒍 𝑻 𝒎 1) Law of length: 𝒏 𝜶 𝟏 𝑻 2) Law of tension: 𝒏 𝜶 𝑻 3) Law of linear density: 𝒏 𝜶 𝟏 𝒎 Linear density = mass per unit length = volume per unit length × density = 𝝅𝒓𝟐𝒍 𝒍 𝝆 𝑨𝒔 𝒏 𝜶 𝟏 𝒎 , , if T and l are constant, we get 𝒏 𝜶 𝟏 𝝅𝒓𝟐𝝆 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 • A sonometer consists of a hollow rectangular wooden box called the sound box. • 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. • The same principle is applied in stringed instruments such as the violin, guitar, tanpura etc.
  • 41. 1) Verification of first law of a vibrating string: By measuring length of wire 𝒏𝟏𝒍𝟏 = 𝒏𝟐𝒍𝟐 = 𝒏𝟑𝒍𝟑 =……………..(constant) Finally, n l = constant 𝒊. 𝒆. 𝒏 𝜶 𝟏 𝒍 …………..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 𝑻𝟏, 𝑻𝟐, 𝑻𝟑, ….. correspond to frequencies 𝒏𝟏, 𝒏𝟐, 𝒏𝟑 ,.....etc. So, 𝒏𝟏 𝑻𝟏 , 𝒏𝟐 𝑻𝟐 , 𝒏𝟑 𝑻𝟑 , … … . . = 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕 ∴ 𝒏 𝑻 = 𝒄𝒐𝒏𝒔𝒕𝒔𝒏𝒕 ∴ 𝒏 𝜶 𝑻 3) Verification of third law of a vibrating string: two wires having different masses per unit lengths 𝒎𝟏 and 𝒎𝟐 (linear densities) ∴ 𝒍𝟏 𝒎𝟏 = 𝒍𝟐 𝒎𝟐 ∴ 𝒍 𝒎 = 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕 ∴ 𝒏 ∝ 𝟏 𝒎
  • 42. Beats • This is an interesting phenomenon based on the principle of superposition of waves. • 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. • Waxing The occurrences of maximum intensity are called waxing • Waning The occurrences of minimum intensity are called waning NOTE: • One waxing and successive waning together constitute one beat. • 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 𝒏𝟏 and 𝒏𝟐. Let us assume for simplicity that the listener is at x = 0. According to the principle of superposition of waves, 𝒚 = 𝒚𝟏 + 𝒚𝟐 𝒚 = 𝑨 𝐬𝐢𝐧(𝟐 𝝅 𝒏 𝒕) 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. • For maximum amplitude (waxing) • For minimum amplitude, A = 0
  • 44. • For maximum amplitude (waxing), A = ±𝑨 𝒕 = 𝟎, 𝟏 𝒏𝟏−𝒏𝟐 , 𝟐 𝒏𝟏−𝒏𝟐 , 𝟑 𝒏𝟏−𝒏𝟐 , … … … … . . Thus, the time interval between two successive maxima of sound is always 𝟐 𝒏𝟏−𝒏𝟐 . Period of beats, T = 𝟐 𝒏𝟏−𝒏𝟐 The number of waxing heard per second is the reciprocal of period of waxing. ∴ frequency of beats, N = 𝒏𝟏 − 𝒏𝟐 • For minimum amplitude (waning), A = 𝟎 𝒕 = 𝟏 𝟐(𝒏𝟏−𝒏𝟐) , 𝟑 𝟐(𝒏𝟏−𝒏𝟐) , 𝟓 𝟐(𝒏𝟏−𝒏𝟐) , … … … … . . Thus, the time interval between two successive minima of sound is 𝟏 𝒏𝟏−𝒏𝟐 . Period of beats, T = 𝟐 𝒏𝟏−𝒏𝟐
  • 45. Applications of beats 1) The phenomenon of beats is used for matching the frequencies of different musical instruments by artists. 2) The speed of an airplane can be determined by using Doppler RADAR. 3) Unknown frequency of a sound note can be determined by using the phenomenon of beats.
  • 46. Characteristics of Sound 1. Loudness: • Loudness is the human perception to intensity of sound. • 1 decibel = 1/10 bel • Loudness is different at different frequencies, even for the same intensity. • For measuring loudness the unit phon is used. Phon is a measure of loudness. 2. Pitch: • It is a sensation of sound which helps the listener to distinguish between a high frequency and a low frequency note. • Pitch is the human perception to frequency- higher frequency denotes higher pitch. • The pitch of a female voice is higher than that of a male voice. 3. Quality or timbre: • Quality of sound is that characteristic which enables us to distinguish between two sounds of same pitch and loudness. • 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.