Physics-02 (Keph 201504) - CIET
Transcript of Physics-02 (Keph 201504) - CIET
Physics-02 (Keph_201504)
Physics 2019 Physics-02 (Keph_201504)Oscillations and Waves
1. Details of Module and its structure
Module Detail
Subject Name Physics
Course Name Physics 02 (Physics part 2,Class XI)
Module Name/Title Unit 10, Module 12, Stationary Waves in Strings
Chapter 15, Waves
Module Id keph_201504_eContent
Pre-requisites Knowledge of wave motion, plane progressive waves, properties of
waves, reflection of sound waves at rigid and non-rigid boundaries and
principle of superposition of waves.
Objectives After going through this module, the learners will be able to:
Recognize that a travelling wave undergoes a phase change of Ο on
reflection from a rigid boundary
Understand the conditions for formation of stationary/standing waves
Recognize the βnodesβ and the βantinodesβ in a stationary wave
Describe formation of standing waves in strings fixed at both the
ends, and stationary waves in pipes
Differentiate and establish a relation between fundamental mode
and overtones
Understand the relation between frequency and length of a given
wire under constant tension using sonometer
Keywords Standing waves, nodes, antinodes, harmonics, sonometer, overtones,
standing waves in strings , standing waves in air coloumns
2. Development Team
Role Name Affiliation
National MOOC
Coordinator (NMC)
Prof. Amarendra P. Behera Central Institute of Educational
Technology, NCERT, New Delhi
Course Coordinator
/ PI
Anuradha Mathur Central Institute of Educational
Technology, NCERT, New Delhi
Subject Matter
Expert (SME)
Vandita Shukla Kulachi Hansraj Model School
Ashok Vihar, New Delhi
Review Team Associate Prof. N.K. Sehgal
(Retd.)
Prof. V. B. Bhatia (Retd.)
Prof. B. K. Sharma (Retd.)
Delhi University
Delhi University
DESM, NCERT, New Delhi
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Physics 2019 Physics-02 (Keph_201504)Oscillations and Waves
TABLE OF CONTENTS
1. Unit Syllabus
2. Module-wise distribution of unit syllabus
3. Words you must know
4. Introduction
5. Standing waves
6. Characteristics of stationary waves
7. Formation of standing waves across a string fixed at both ends
8. Graphical representation of stationary wave in a string
9. Analytical treatment of standing waves
10. Fundamental mode and overtone
11. Sonometer
12. Summary
1. UNIT SYLLABUS
Unit: 10
Oscillations and Waves
Chapter 14: oscillations
Periodic motion, time period, frequency, displacement as a function of time , periodic
functions Simple harmonic motion (S.H.M) and its equation; phase; oscillations of a loaded
spring-restoring force and force constant; energy in S.H.M. Kinetic and potential energies;
simple pendulum derivation of expression for its time period.
Free forced and damped oscillations (qualitative ideas only) resonance
Chapter 15 Waves
Wave motion transverse and longitudinal waves, speed of wave motion, displacement, relation
for a progressive wave, principle of superposition of waves, reflection of waves, standing waves
in strings and organ pipes, fundamental mode and harmonics, beats, Doppler effect
2. MODULE-WISE DISTRIBUTION OF UNIT SYLLABUS 15 MODULES
Module 1
Periodic motion
Special vocabulary
Time period, frequency,
Periodically repeating its path
Periodically moving back and forth about a point
Mechanical and non-mechanical periodic physical
quantities
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Module 2 Simple harmonic motion
Ideal simple harmonic oscillator
Amplitude
Comparing periodic motions phase,
Phase difference
Out of phase
In phase
not in phase
Module 3
Kinematics of an oscillator
Equation of motion
Using a periodic function (sine and cosine functions)
Relating periodic motion of a body revolving in a circular
path of fixed radius and an Oscillator in SHM
Module 4
Using graphs to understand kinematics of SHM
Kinetic energy and potential energy graphs of an oscillator
Understanding the relevance of mean position
Equation of the graph
Reasons why it is parabolic
Module 5
Oscillations of a loaded spring
Reasons for oscillation
Dynamics of an oscillator
Restoring force
Spring constant
Periodic time spring factor and inertia factor
Module 6
Simple pendulum
Oscillating pendulum
Expression for time period of a pendulum
Time period and effective length of the pendulum
Calculation of acceleration due to gravity
Factors effecting the periodic time of a pendulum
Pendulums as βtime keepersβ and challenges
To study dissipation of energy of a simple pendulum by
plotting a graph between square of amplitude and time
Module 7 Using a simple pendulum plot its L-T2graph and use it to
find the effective length of a secondβs pendulum
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To study variation of time period of a simple pendulum of a
given length by taking bobs of same size but different masses
and interpret the result
Using a simple pendulum plot its L-T2graph and use it to
calculate the acceleration due to gravity at a particular place
Module 8
Free vibration natural frequency
Forced vibration
Resonance
To show resonance using a sonometer
To show resonance of sound in air at room temperature
using a resonance tube apparatus
Examples of resonance around us
Module 9
Energy of oscillating source, vibrating source
Propagation of energy
Waves and wave motion
Mechanical and electromagnetic waves
Transverse and longitudinal waves
Speed of waves
Module 10 Displacement relation for a progressive wave
Wave equation
Superposition of waves
Module 11
Properties of waves
Reflection
Reflection of mechanical wave at i)rigid and ii)non-rigid
boundary
Refraction of waves
Diffraction
Module 12
Special cases of superposition of waves
Standing waves
Nodes and antinodes
Standing waves in strings
Fundamental and overtones
Relation between fundamental mode and overtone
frequencies, harmonics
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To study the relation between frequency and length of a
given wire under constant tension using sonometer
To study the relation between the length of a given wire and
tension for constant frequency using a sonometer
Module13 Standing waves in pipes closed at one end,
Standing waves in pipes open at both ends
Fundamental and overtones
Relation between fundamental mode and overtone
frequencies
Harmonics
Module 14 Beats
Beat frequency
Frequency of beat
Application of beats
Module 15
Doppler effect
Application of Doppler effect
MODULE 12
3. WORDS YOU MUST KNOW
Let us remember the words we have been using in our study of this physics course
Wave motion: method of energy transfer from a vibrating source to any observer.
Mechanical wave energy transfer by vibration of material particles in response to a
vibrating source examples water waves, sound waves , waves in strings
The speed of wave in medium depends upon elasticity and density
Longitudinal mechanical wave a wave in which the particles of the medium vibrate
along the direction of propagation of the wave
Transverse mechanical wave a wave in which the particles of the medium vibrate
perpendicular to the direction of propagation of the wave
A progressive wave: The propagation of a wave in a medium means the particles of the
medium perform simple harmonic motion without moving from their positions, then the
wave is called a simple harmonic progressive wave
Displacement relation for a Progressive wave: The displacement of the particle at an
instant t is given by,
π¦ = π sin (π π‘ β π ) OR π¦ = π sin (π π‘ β π π₯ )
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If Ο be the phase difference between the above wave propagating along the +X direction
and another wave, then the equation of that wave will be
π¦ = π sin {2π (π‘
π β
π₯
π) + π}
π¦ = π sin(π π‘ β π π₯ + π)
The displacement could also be expressed in terms of the cosine function without
affecting any of the subsequent relation.
Particle Velocity: The equation of a plane progressive wave propagating in the positive
direction of X-axis is given by
π£ =ππ¦
ππ‘= π π πππ (π π‘ β π π₯)
The maximum particle velocity is given by,
π£πππ₯ = π π, this is known as velocity amplitude of particle.
Particle Acceleration: The instantaneous acceleration π of a particle is
π =ππ’
ππ‘ = π2 π sin (π π‘ β π π₯) = βπ2 π¦
The maximum value of the particle displacement y is a. Therefore, acceleration
amplitude is ππππ₯ = β π2 π
Principle of superposition: The net displacement of the medium / particles (through
which waves travel) due to the superposition is equal to the sum of individual
displacements (produced by each wave).
Progressive wave: In progressive wave, the disturbance produced in the medium travels
onward, it being handed over from one particle to the next. Each particle executes the
same type of vibration as the preceding one, though not at the same time. In this wave,
energy propagates from one point in space to the other.
Wave properties waves show properties of reflection, refraction, superposition
(interference, stationary waves, beats), diffraction and polarization
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4. INTRODUCTION
We have learnt the basic properties of waves. Waves are generated by a vibrating source.
Raindrops making waves on water surface, the superposition principle is difficult to apply.
The raindrop pushes a section of water down which in turn oscillates for a short duration
due to inertia and elasticity. You can experience a pulse on water surface by dropping a
water drop from your wet hands on a still water surface in a bucket of water.
The waves pass crossing each other without being disturbed. The net displacement of the
medium at any point in space or time is simply the sum of the individual wave
displacements. Hence the result is not predictable.
A wave in which energy is transferred from one place to another as a result of its propagation is
called a progressive wave. An ultraviolet light wave which transfers energy from the sun to the
earth for instance is a progressive wave.
In general, waves that move from one point to another transfer some kind of energy. In a
progressive wave, the shape of the wave itself, is what gets transferred not the actual components
of the medium
Sometimes on vibrating a string, or cord, or chain, or cable you must have felt that it's possible to
get it to vibrate in a manner such that you're generating a wave, but the wave doesn't propagate.
It just sits there vibrating up and down in localized place.
Such a wave is called a standing wave.
Why is it called standing wave?
This module deals with the formation of these types of waves.
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5. STANDING WAVES
When two identical waves of the same amplitude and frequency travel in opposite directions
with the same speed along the same path superpose each other, the resultant wave does not travel
in the either direction and is called a stationary or standing wave. It is called a standing wave
because it does not appear to move.
We will see in this module that standing wave can also be created when a single wave is
reflected off a fixed boundary (string reflecting with one end of the string attached to the wall).
Standing wave
In stationary or standing waves, the shape or profile of the wave stays fixed in a medium.
An example of a stationary wave is the wave produced on the string of a string instrument.
When the string is plucked, a wave is caused to travel up and down.
Since both ends of the string are fixed, the waveform is reflected back up and down the
string or along its path. This confines a wave to stay within it.
The basic characteristic of a progressive wave is propagation of energy through the medium. In a
stationary βwaveβ the energy does not travel or propagate forward. Then why should we call it a
wave?
Watch how standing waves are formed at
http://www.walter-fendt.de/ph14e/stwaverefl.htm
http://www.physicsclassroom.com/mmedia/waves/swf.cfm
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Types of stationary waves
1. Transverse stationary waves: when two identical transverse waves travelling in
opposite direction overlap, a transverse standing wave is formed. For example, transverse
standing waves are formed in sonometer experiment.
In this module we will learn about the transverse standing waves in detail.
2. Longitudinal stationary waves: when two identical longitudinal waves travelling in
opposite direction overlap, a longitudinal standing wave is formed. For example,
longitudinal standing waves are formed in resonance experiment and organ pipes.
6. CHARACTERISTICS OF STATIONARY WAVES
In stationary wave, the disturbance does not advance forward.
In stationary waves, there are certain points called nodes where the particles are
permanently at rest and certain other points called antinodes where the particles vibrate
with maximum amplitude. The nodes and antinodes are formed alternately.
All the particles of the medium except those at the nodes, execute simple harmonic motion
with the same time period about their mean positions.
The amplitude of vibration increases gradually from zero to maximum from a node to an
antinode.
During the formation of standing waves the medium is split up into segments. The
particles in a segment vibrate in phase. The particles in one segment are out of phase with
the particles in the neighboring segment by 180o.
The distance between two successive nodes or antinodes is Ξ»/2
In a given segment, the particles attain their maximum or minimum velocity and
acceleration at the same instant.
There is no net transport of energy in the medium.
Compressions and rarefactions do not travel forward as in progressive waves. They appear
and disappear alternately, at the same place.
During each vibration, all the particles pass simultaneously through their mean positions
twice, with maximum velocity which is different for different particles.
COMPARISION BETWEEN STANDING WAVES AND TRAVELING WAVES
While traveling waves transmit energy a standing wave does not. However, there is
energy associated with standing waves.
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Any given point on a traveling wave will have amplitudes ranging from the minimum to
the maximum amplitude, i.e. all point can attain any amplitude. Whereas a given point on
a standing wave will have amplitudes ranging from a max to min, but not necessarily the
max and min of the wave.
The wavelength of a traveling wave is the physical distance from a peak to the next
peak or from a trough to the next trough. The wavelength of a standing wave is
twice the distance between nodes or twice the distance between antinodes.
7. FORMATION OF STANDING WAVES ACROSS A STRING FIXED AT BOTH
ENDS
In a string, a wave going to the right will get reflected at one end, which in turn will travel and
get reflected from the other end. This will go on until there is a steady wave pattern set up on the
string. Such wave patterns are called standing waves or stationary waves.
STATIONARY (OR STANDING) WAVES
When two identical transverse or longitudinal, progressive waves propagate in a bounded medium
with the same speed, but in opposite directions, then by their superposition, a new type of wave is
produced which appears stationary in the medium. This wave is called the βstationary (or standing)
waveβ.
For example,
A rubber band held at its ends is made to vibrate
http://cdn.playbuzz.com/cdn/96f59b8c-d770-410c-8572-ba796ee89bb3/230173a3-67cf-41cc-
83a4-77977980787c.jpg
You can observe the nodes and antinodes in a rubber band stretched between our fingers
and thumb. On vibrating the rubber band the above discussion would become clearer.
When a wave is sent along a string, it is reflected from the end of the string; then reflected and the
incident waves superpose to form stationary waves in the string.
Transverse stationary waves are formed in the string of sitar, violin, guitar, etc. Similarly, a
longitudinal wave sent in an air column of a pipe is reflected from the end of the pipe, and
the reflected and incident waves superpose to form stationary waves in the air column.
Longitudinal stationary waves are formed in the air-columns of flute, bigule, bina, whistle,
etc.
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NODES AND ANTINODES
The characteristic of the stationary wave is that some particles of the medium remain permanently
at rest, while some other particles undergo maximum displacement compared to others. The former
are called the βnodesβ and the latter the βantinodesβ. The points at which the amplitude is zero
(i.e., where there is no motion at all) are nodes; the points at which the amplitude is the largest are
called antinodes.
Condition of Formation of Stationary Waves:
For the formation of stationary waves, the medium should have a boundary.
The wave propagating on such a medium will be reflected at the boundary and produce a
wave of the same kind travelling in the opposite direction.
The superposition of the two waves will give rise to a stationary wave.
Hence, a βboundedβ medium is an essential condition for the formation of stationary waves
8. GRAPHICAL REPRESENTATION OF STATIONARY WAVE IN A STRING
Stationary waves are formed due to superposition of a wave and its reflection from a rigid
Surface.
In the diagram, the blue line shows the resultant wave obtained by taking the algebraic sum of
the displacements of the two waves at every point which were travelling in the opposite
directions.
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Stationary waves arising from superposition of two harmonic waves travelling in opposite
directions.
Note that the positions of zero displacement (nodes) remain fixed at all times.
If we draw the resultant wave at different instants in the same diagram below, then the
nature of the wave becomes clearer. It is seen in the diagram that the wave does not advance
towards right or left, but undergoes expansion and contraction, remaining stationary in its
position.
In longitudinal waves, the particles of the medium are displaced in the direction of the
wave. At any instant, the particles on the two sides of a node move in opposite directions.
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When they move away from the node, then the pressure at the node decreases; and when they
move towards the node, then the pressure at the node increases.
Thus, the change in pressure in maximum is at nodes.
On the other hand, the particles of the two sides of an antinode move in the same direction at any
instant. Hence, there is no change in pressure at the antinodes.
9. ANALYTICAL TREATMENT OF STANDING WAVES
Let
π² = π π¬π’π§ (ππ β π€π±)
be the equation of an incident progressive wave at any instant π‘ then, the equation of the wave
reflected from the closed end will be
π² = π π¬π’π§[ππ + π€π± + π],
because the wave reflected from the closed end suffers a phase change of π ππ πππΒ°.
The equations of these waves at different times (π‘ = 0, π/4, π/2,3π/4, π) will be as shown in
the following table:
Time Equation of Incident
Wave
Equation of Wave Reflected
π¨ π π = π π¦ = π sin (π π‘ β π π₯) π¦ = π sin [π π‘ + π π₯ + π]
π¨ π π = π π¦ = βπ sin π π₯
π¦ = βπ sin π π₯ (π πππ ππ ππππππππ‘)
π¨ π π = π»
π
π¦ = +π cos π π₯ π¦ = β a cos π π₯ (πππππ ππ‘π π‘π ππππππππ‘)
π¨ π π = π»/π π¦ = + π sin π π₯
π¦ = +π sin π π₯ (π πππ ππ ππππππππ‘ )
π¨ π π = ππ»
π
π¦ = βπ cos π π₯ π¦ = +π cos π π₯ (πππππ ππ‘π π‘π ππππππππ‘)
π¨ π π = π» π¦ = βπ sin π π₯ π¦ = β π sin π π₯ (π πππ ππ ππππππππ‘)
The formation of stationary waves as a result of reflection of progressive waves from a close end
(or rigid surface).
Suppose, the incident wave is going from left to right, and the wave reflected from the close end
is coming back from right to left.
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In the figure, the incident and the reflected waves are shown by continuous and dotted thin lines
respectively.
We considered above reflection at one boundary. But there are familiar situations (a string fixed
at either end or an air column in a pipe with either end closed) in which reflection takes place at
two or more boundaries. In a string, for example, a wave going to the right will get reflected at
one end, which in turn will travel and get reflected from the other end.
This will go on until there is a steady wave pattern set up on the string. Such wave
patterns are called standing waves or stationary waves. To see this mathematically, consider a
wave travelling along the positive direction of x-axis and a reflected wave of the same amplitude
and wavelength in the negative direction of x-axis.
Consider a wave travelling along the positive direction of x-axis and a reflected wave of the same
amplitude and wavelength in the negative direction of x-axis.
The wave travelling along positive direction of x-axis can be represented as
ππ(π, π) = π π¬π’π§(ππ β ππ)
The wave travelling along negative direction of x-axis can be represented as
ππ(π, π) = π π¬π’π§[ππ + π€π± + π] = βπ πππ(ππ + ππ)
The resultant wave on the string is, according to the principle of superposition:
π¦(π₯, π‘) = π¦1(π₯, π‘) + π¦2(π₯, π‘)
= π[π ππ(ππ‘ β ππ₯) β sin( ππ‘ + ππ₯)]
Using the familiar trigonometric identity
ππ’π§(π β π) β ππ’π§(π + π) = βπππ¨π¬ππ¬π’π§π
We get,
A = ππ‘ and B =kx
π²(π±, π) = β(ππ π¬π’π§ π€π±) ππ¨π¬ ππ
Note: We can consider any of equations
π(π, π) = π π¬π’π§(ππ β ππ)
or
π(π, π) = π π¬π’π§(ππ β ππ)
There is a phase change of Ο i.e. the initial phase of y = π π¬π’π§(ππ β ππ) is Ο.
π(π, π) = π π¬π’π§(ππ β ππ)
π(π, π) = π π¬π’π§(ππ β ππ) = βπ π¬π’π§(ππ β ππ) = π π¬π’π§(ππ β ππ + π )
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Both the equations are progressive waves, so we can take any of these equations.
This equation represents standing wave.
The amplitude of this wave is ππ πππππ.
Thus in this wave pattern,
The amplitude varies from point to point, but each element of the string oscillates
with the same angular frequency Ο or time period.
There is no phase difference between oscillations of different elements of the wave.
The string as a whole vibrates in phase with different amplitudes at different points.
The wave pattern is neither moving to the right nor to the left.
Hence they are called standing or stationary waves.
MODES OF OSCILLATIONS
The most significant feature of stationary waves is that the boundary conditions constrain the
possible wavelengths or frequencies of vibration of the system.
The system cannot oscillate with any arbitrary frequency (contrast this with a harmonic
travelling wave), but is characterized by a set of natural frequencies or normal modes of
oscillation.
Let us determine these normal modes for a stretched string fixed at both ends.
In string instruments the string is fixed between two fixed points mounted on a hollow
wooden box, the string is plucked to vibrate. Sounds of different frequencies are produced
depending upon the length, thickness and material of the string.
POSITION OF NODES
Nodes are the points on the string where the amplitude of oscillation of constituents is zero.
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From equation,
π(π, π) = β(ππ πππππ) πππ ππ
The positions of nodes (where the amplitude is zero) are given by
πππ¬π’π§π€π± = π
π ππ ππ₯ = 0
which implies
ππ = ππ ; π = π, π, π, π, β¦.
Since π = ππ /π,
we get
π =ππ
π; π = π, π, π, π, β¦.
Clearly, the distance between any two successive nodes is π
π.
POSITION OF ANTINODES
Antinodes are the points on the string where the amplitude of oscillation of constituents is
maximum.
In the same way, the positions of antinodes (where the amplitude is the largest) are given by the
largest value of π ππ ππ₯:
πππ¬π’π§ π€π± = π
which implies
ππ = (π + Β½) π ; π = π, π, π, π, . ..
With π = ππ /π, we get
π = (π + Β½)π
π ; π = π, π, π, π, β¦ ..
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Again the distance between any two consecutive antinodes is π
π.
If the length of the string is πΏ, then its first end can be taken as π₯ = 0 while the other end is denoted
as π₯ = πΏ
Eq. π₯ =ππ
2; π = 0, 1, 2, 3, . .. can be applied to the case of a stretched string of length πΏ fixed at
both ends.
Taking one end to be at π₯ = 0, the boundary conditions are that π = π and π = π³ are positions
of nodes.
The π = π condition is already satisfied.
The π = π³ node condition requires that the length L is related to Ξ» by
π³ = π π/π ; π = π, π, π, . ..
Thus, the possible wavelengths of stationary waves are constrained by the relation
Ξ» = ππ³
π; π = π, π, π, β¦
with corresponding
frequencies can be obtained by using relation
π = π/ π
where π is the speed of wave in the given medium.
π = ππ/ππ³, πππ π = π, π, π,
We have thus obtained the natural frequencies - the normal modes of oscillation of the system.
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10. FUNDAMENTAL MODE AND OVERTONE
Any system, in which standing waves can form, has numerous natural frequencies.
These are called overtones. If the frequency of overtone is a multiple of the fundamental
frequency, it is called a harmonic.
The set of all possible standing waves are known as the harmonics of a system. The simplest of
the harmonics is called the fundamental or first harmonic. Subsequent standing waves are called
the second harmonic, third harmonic, etc. The harmonics above the fundamental, especially in
music theory, are sometimes also called overtones.
RELATION BETWEEN FUNDAMENTAL MODE AND OVERTONE
n =1
The lowest possible natural frequency of a system is called its fundamental mode or the first
harmonic.
For the stretched string, fixed at either end (corresponding to n = 1), it is given by
ππ = π―/ππ
Here π£ is the speed of wave determined by the properties of the medium.
The speed of a wave in a string π = βπ
π
T= tension,
ΞΌ = mass per unit length
Thus
ππ =π
ππβ
π
π
The n = 2 frequency is called the second harmonic or first overtone;
π2 =2π£
2πΏ
π2 = 2π1
π§ = π is the third harmonic or second overtone
ππ = ππ―/ππ
ππ = π ππ
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Hence,
ππ = π π1; π = 1,2,3 β¦ β¦ ..
Fig. shows the first six harmonics of a stretched string fixed at either end.
A string need not vibrate in one of these modes only. Generally, the vibration of a string will be a
superposition of different modes; some modes may be more strongly excited and some less.
Musical instruments like sitar or violin are based on this principle. Where the string is plucked or
bowed, determines which modes are more prominent than others.
The following link shows the standing waves and harmonics
https://youtu.be/Ew0fZh9INbQ
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First six harmonics notice the length of the string, and the material remains the same in
each case.
EXAMPLE
A string clamped at both its ends is stretched out, it is then made to vibrate in its
fundamental mode at a frequency of 45 π»π§.
The linear mass density of the string is 4.0 Γ 10-2 kg / m and its mass is 2 Γ 10β2 ππ.
Calculate:
( π ) the velocity of a transverse wave on the string,
( ππ ) the tension in the string.
SOLUTION:
Mass of the string, π = 2 Γ 10β2 ππ
Linear density of the string Β΅ = 4 Γ 10β2 ππ/π
Frequency, π1 = 45 π»π§
Using Linear density of the string Β΅ = mass / length
We know, length of the wire = (2 Γ 10β2)/(4 Γ 10β2) = 0.5 π
as, π = 2π/π
For fundamental node, π = 1 => π = 2π = 2 Γ 0.5 = 1π
(i) Therefore, speed of the transverse wave,
Speed = frequency x wavelength
= 1 Γ 45 = 45 π/π
(ii) π£ = βπ
π
Tension in the string = Β΅π£2
= 4 Γ 10β2 Γ 452 = ππ π΅
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EXAMPLE
A steel bar of length 200 ππ is nailed at its midβpoint. The fundamental frequency of
longitudinal vibrations of the rod is 2.53 ππ»π§. At what speed will the sound be able to
travel through steel?
SOLUTION
Length, π = 200ππ = 2π
Fundamental frequency of vibration,
π = 2.53 ππ»π§ = 2.53 Γ 103π»π§
The bar is then plucked at its mid-point, forming an antinode (A) at its center, and nodes (N) at
its two edges, as shown in the figure below:
As, the distance between two successive nodes is π/2 => π = π/2
or, π = 2 Γ 2 = 4 π
Thus, sound travels through steel at a speed of
π£ = ππ
π£ = 4 Γ 2.53 Γ 103 = ππ. ππ ππ/π
EXAMPLE
The transverse displacement of a wire (clamped at both its ends) is described as:
y (x, t) = 0.06sin(2Ο3x)cos(120Οt)
The mass of the wire is 6 x 10-2 kg and its length is 3m.
Provide answers to the following questions:
(i) Is the function describing a stationary wave or a travelling wave?
(ii) Interpret the wave as a superposition of two waves travelling in opposite directions.
Find the speed, wavelength and frequency of each wave.
(iii) Calculate the wireβs tension.
[x and y are in meters and t in secs]
SOLUTION: The general equation of a stationary wave is:
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π¦(π₯, π‘) = 2π π ππ ππ₯ πππ ππ‘
Comparing π¦(π₯, π‘) = 0.06π ππ(2π3π₯)πππ (120ππ‘) with the general equation.
(i) The given function describes a stationary wave.
(ii) The transverse displacement of the wires is described as:
0.06π ππ(2π3π₯)πππ (120ππ‘)
Comparing with general, we get:
2π/π = 2π/3
wavelength π = ππ
Also, 2ππ£/π = 120π, speed π£ = πππ π/π
And, πΉππππ’ππππ¦ = π£/π = 180/3 = ππ π―π
(iii) Velocity of the transverse wave, π£ = 180π/π
The stringβs mass, π = 6 Γ 10β2 ππ
String length, π = 3 π
Mass per unit length of the string, π = π/π = (6 Γ 10β2)/3 = 2 Γ 10β2ππ/π
Let the tension in the wire be π
Therefore, π = π£2π
= 1802 Γ 2 Γ 10β2
= πππ π΅
11. SONOMETER
Sonometer consists of a hollow rectangular wooden box of more than one-meter length, with a
hook at one end and a pulley at the other end. One end of a string is fixed at the hook and the
other end passes over the pulley. A hanger with weights is attached to the free end of the string.
Two adjustable wooden bridges are put over the board, so that the length of string can be
adjusted.
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Production of transverse waves in stretched strings
If a string which is stretched between two fixed points is plucked at its center, vibrations
produced and it move out in opposite directions along the string. Because of this, a transverse
wave travels along the string.
If a string of length Ζ having mass per unit length m is stretched with a tension T, the
fundamental frequency of vibration π is given by;
π =1
2πΏβ
π
π
Laws of transverse vibrations on a stretched string
Law of Length: The frequency of vibration of a stretched string varies inversely as its
resonating length (provided its mass per unit length and tension remain constant).
π β1
π
Law of Tension: The frequency of vibration of a stretched string varies directly as the
square root of its tension, (provided its resonating length and mass per unit length of the
wire remains constant).
π β βπ
Relation between frequency and length
From the law of length, f Γ Ζ = constant
A graph between π and 1/π will be a straight line.
Relation between length and tension
From the equation for frequency, βT / Ζ = constant.
A graph between π and π2 will be a straight line.
The following video shows the experiment which is done to describe the relation between
frequency and length of a given wire under constant tension using sonometer.
https://youtu.be/QXJ2oc4hx98
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EXPERIMENTS IN THE LABORATORY
(i) To study the relation between frequency and length of a given wire under constant
tension using a sonometer.
(ii) To study the relation between the length of a given wire and tension for constant
frequency using a sonometer.
APPARATUS AND MATERIAL REQUIRED:
Six tuning forks of known frequencies, sonometer, meter scale, rubber pad, paper rider, hanger
with half kilogram weights, wooden bridges.
SONOMETER
It consists of a long sounding board or a hollow wooden box with a peg G at one end and a
pulley at the other end as shown in Figure
One end of a metal wire S is attached to the peg and the other end passes over the pulley P. A
hanger H is suspended from the free end of the wire. By placing slotted weights on the hanger,
tension is applied to the wire. By placing two bridges A and B under the wire, the length of the
vibrating wire can be fixed. Position of one of the bridges say bridge A is kept fixed so that by
varying the position of other bridge, say bridge B, the vibrating length can be altered.
The function of the wooden box is to create stationary waves in the enclosed air,
Together the vibrating string and the air in the wooden make the sound produced
by the string louder or this increases the intensity of sound
The wooden box has holes. The holes bring the inside air in contact with the outside
air.
C is a hook where one end of the wire is fixed.
P is a friction less pulley, the sonometer wire is made to pass over it
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H is a hanger with slotted weights. the weights provide the tension to the wire. the
tension can be changed using suitable weights
The stationary wave is formed between the wedges A and B. both A and B
becoming nodes for the wave (as shown in the figure). Wedges usually have a thin
metal strip embedded in the wooden prism shaped blocks.
A scale is appropriately attached to the sonometer to facilitate the measurement
between A and B
PRINCIPLE
(a)The frequency f of the fundamental mode of vibration of a string is given by
π =π
ππ₯β
π
π¦
Here, m = mass per unit length of the string;
π = length of the string between the wedges;
T = Tension in the string (including the weight of the hanger) = Mg
M = mass suspended, including the mass of the hanger
(a) For a given m and fixed T,
π βπ
π ππ π π = ππππππππ
(b) If frequency f is constant, for a given wire (m is constant), βπ»
π is constant. That is ππ β T
Variation of resonant length with frequency of tuning fork
(i) VARIATION OF FREQUENCY WITH LENGTH
PROCEDURE
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Set up the sonometer on the table and clean the groove on the pulley to ensure that it has
minimum friction. Stretch the wire by placing a suitable load on the hanger.
Why is load added to the hanger?
Why should the pulley be frictionless?
Why should the wire be of uniform area of cross section?
Should the wire be of homogeneous material?
Set a tuning fork of frequency π1 into vibrations by striking it against the rubber pad and
hold it near one of your ears. Pluck the sonometer wire and compare the two sounds, one
produced by the tuning fork and the other by the plucked wire. Make a note of
difference between the two sounds.
Why will the sounds be different?
When will they be the same?
Will you be able to distinguish which sound comes from the tuning fork?
Adjust the vibrating length of the wire by sliding the bridge B till the two sounds appear
alike.
Can this be done by shifting A?
For final adjustment, place a small paper rider R in the middle of wire AB. Sound the
tuning fork and place its shank stem on the sonometer box. Slowly adjust the position of
bridge B till the paper rider is agitated violently, which indicates resonance. The length
of the wire between A and B is the resonant length such that its frequency of vibration
of the fundamental mode equals the frequency of the tuning fork. Measure this length
with the help of a metre scale.
Repeat the above procedures for other five tuning forks keeping the load on the hanger
unchanged. Plot a graph between n and Ζ
Why repeat the readings?
Think of the sources of error
After calculating frequency, n of each tuning fork, plot a graph between n and 1/Ζ where
Ζ is the resonating length.
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Variation of 1/l with frequency (n)
OBSERVATIONS (A)
Tension (constant) on the wire (weight suspended from the hanger including its own weight) T =
... N
Variation of frequency with length
CALCULATIONS AND GRAPH
Calculate the product n Ζ for each fork and, calculate the reciprocals, 1
π of the resonating lengths Ζ.
Plot 1
π vs n, taking n along x axis and
1
π along y axis, starting from zero on both axes. See
whether a straight line can be drawn from the origin to lie evenly between the plotted points.
RESULT
Check if the product n Ζ is found to be constant and the graph of π
π vs n is also a straight
line.
Therefore, for a given tension, the resonant length of a given stretched string varies as
reciprocal of the frequency
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THINK ABOUT THESE
Error may occur in measurement of length Ζ. There is always an uncertainty in
setting the bridge in the final adjustment.
Some friction might be present at the pulley and hence the tension may be less than
that actually applied.
The wire may not be of uniform cross section.
(ii) VARIATION OF RESONANT LENGTH WITH TENSION FOR CONSTANT
FREQUENCY
Select a tuning fork of a certain frequency (say 256 Hz) and hang a load of 1kg from the
hanger. Find the resonant length as before
Increase the load on the hanger in steps of 0.5 kg and each time find the resonating
length with the same tuning fork. Do it for at least four loads.
Record your observations.
Plot graph between π2 and T as shown
Graph between l2 and T
OBSERVATIONS (B)
Frequency of the tuning fork = ... Hz
Variation of resonant length with tension
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CALCULATIONS AND GRAPH
Calculate the value of π/π2 for the tension applied in each case. Alternatively, plot a graph of
π2 vs T, taking π2 along y-axis and T along the x-axis.
RESULT
It is found that value of π»/ππ is constant within experimental error.
The graph of ππ vs T is found to be a straight line.
This shows that π2 Ξ± T or π β βπ.
Thus, the resonating length varies as square root of tension for a given frequency of
vibration of a stretched string.
THINK ABOUT THESE
Pulley should be frictionless ideally. In practice friction at the pulley should be
minimized by applying grease or oil on it.
Wire should be free from kinks and of uniform cross section, ideally. If there are kinks,
they should be removed by stretching as far as possible.
Bridges should be perpendicular to the wire, its height should be adjusted so that a node
is formed at the bridge.
Tuning fork should be vibrated by striking its prongs against a soft rubber pad.
Load should be removed after the experiment.
Error may occur in measurement of length Ζ. There is always an uncertainty in setting the
bridge in the final adjustment.
Some friction might be present at the pulley and hence the tension may be less than that
actually applied.
The wire may not be of uniform cross section.
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Care should be taken to hold the tuning fork by the shank only.
You could also do the following
Is the principle of superposition of waves satisfied in the apparatus?
Where are the stationary waves formed?
Why are stationary waves formed?
Identify the nodes and antinodes in the string of your sonometer.
What is the ratio of the first three harmonics produced in a stretched string fixed
at two ends?
Keeping material of wire and tension fixed, how and why will the resonant length
change if the diameter of the wire is increased?
SUGGESTED ACTIVITIES
Take wires of the same material but of three different diameters and find the value of Ζ for each
of these for a given frequency, n and tension, T.
Plot a graph between the values of m and π
ππ obtained, in 1 above, with m along X axis.
Pluck the string of a stringed musical instrument like a sitar, violin or guitar with different
lengths of string for same tension or same length of string with different tension. Observe how
the frequency of the sound changes.
12. SUMMARY
The interference of two identical waves moving in opposite directions
produces standing waves. For a string with fixed ends, the standing wave is given
by π¦(π₯, π‘) = [2ππ ππ ππ₯ ] πππ ππ‘
Standing waves are characterized by fixed locations of zero displacement
called nodes and fixed locations of maximum displacements called antinodes. The
separation between two consecutive nodes or antinodes is Ξ»/2.
A stretched string of length L fixed at both the ends vibrates with frequencies given
by 1
2
π£
2πΏ n = 1, 2, 3, ...
The set of frequencies given by the above relation are called the normal modes of
oscillation of the system. The oscillation mode with lowest frequency is called
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the fundamental mode or the first harmonic. The second harmonic is the oscillation
mode with n = 2 and so on.