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 π on

reflection from a rigid boundary

Understand the conditions for formation of stationary/standing waves

Recognize the ‘nodes’ and the ‘antinodes’ 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



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



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 ‘time keepers’ 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 second’s pendulum



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



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,

𝑦 = 𝑎 sin (𝜔 𝑡 − 𝜙 ) OR 𝑦 = 𝑎 sin (𝜔 𝑡 − 𝑘 𝑥 )



If ϕ be the phase difference between the above wave propagating along the +X direction

and another wave, then the equation of that wave will be

𝑦 = 𝑎 sin {2𝜋 (𝑡

𝑇 −

𝑥

𝜆) + 𝜙}

𝑦 = 𝑎 sin(𝜔 𝑡 − 𝑘 𝑥 + 𝜙)

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

𝑣 =𝑑𝑦

𝑑𝑡= 𝜔 𝑎 𝑐𝑜𝑠 (𝜔 𝑡 − 𝑘 𝑥)

The maximum particle velocity is given by,

𝑣𝑚𝑎𝑥 = 𝜔 𝑎, this is known as velocity amplitude of particle.

Particle Acceleration: The instantaneous acceleration 𝑓 of a particle is

𝑓 =𝑑𝑢

𝑑𝑡 = 𝜔2 𝑎 sin (𝜔 𝑡 − 𝑘 𝑥) = −𝜔2 𝑦

The maximum value of the particle displacement y is a. Therefore, acceleration

amplitude is 𝑓𝑚𝑎𝑥 = − 𝜔2 𝑎

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



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.



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 ‘wave’ 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

http://www.walter-fendt.de/ph14e/stwaverefl.htm

http://www.physicsclassroom.com/mmedia/waves/swf.cfm



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 λ/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.



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 ‘stationary (or standing)

wave’.

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.

http://cdn.playbuzz.com/cdn/96f59b8c-d770-410c-8572-ba796ee89bb3/230173a3-67cf-41cc-83a4-77977980787c.jpg

http://cdn.playbuzz.com/cdn/96f59b8c-d770-410c-8572-ba796ee89bb3/230173a3-67cf-41cc-83a4-77977980787c.jpg



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 “nodes” and the latter the “antinodes”. 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 “bounded” 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.



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.



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

𝐲 = 𝐚 𝐬𝐢𝐧 (𝛚𝐭 − 𝐤𝐱)

be the equation of an incident progressive wave at any instant 𝑡 then, the equation of the wave

reflected from the closed end will be

𝐲 = 𝐚 𝐬𝐢𝐧[𝛚𝐭 + 𝐤𝐱 + 𝛑],

because the wave reflected from the closed end suffers a phase change of 𝝅 𝒐𝒓 𝟏𝟖𝟎°.

The equations of these waves at different times (𝑡 = 0, 𝑇/4, 𝑇/2,3𝑇/4, 𝑇) will be as shown in

the following table:

Time Equation of Incident

Wave

Equation of Wave Reflected

𝑨 𝒕 𝒕 = 𝒕 𝑦 = 𝑎 sin (𝜔 𝑡 − 𝑘 𝑥) 𝑦 = 𝑎 sin [𝜔 𝑡 + 𝑘 𝑥 + 𝜋]

𝑨 𝒕 𝒕 = 𝟎 𝑦 = −𝑎 sin 𝑘 𝑥

𝑦 = −𝑎 sin 𝑘 𝑥 (𝑠𝑎𝑚𝑒 𝑎𝑠 𝑖𝑛𝑐𝑖𝑑𝑒𝑛𝑡)

𝑨 𝒕 𝒕 = 𝑻

𝟒

𝑦 = +𝑎 cos 𝑘 𝑥 𝑦 = − a cos 𝑘 𝑥 (𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑡𝑜 𝑖𝑛𝑐𝑖𝑑𝑒𝑛𝑡)

𝑨 𝒕 𝒕 = 𝑻/𝟐 𝑦 = + 𝑎 sin 𝑘 𝑥

𝑦 = +𝑎 sin 𝑘 𝑥 (𝑠𝑎𝑚𝑒 𝑎𝑠 𝑖𝑛𝑐𝑖𝑑𝑒𝑛𝑡 )

𝑨 𝒕 𝒕 = 𝟑𝑻

𝟒

𝑦 = −𝑎 cos 𝑘 𝑥 𝑦 = +𝑎 cos 𝑘 𝑥 (𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑡𝑜 𝑖𝑛𝑐𝑖𝑑𝑒𝑛𝑡)

𝑨 𝒕 𝒕 = 𝑻 𝑦 = −𝑎 sin 𝑘 𝑥 𝑦 = − 𝑎 sin 𝑘 𝑥 (𝑠𝑎𝑚𝑒 𝑎𝑠 𝑖𝑛𝑐𝑖𝑑𝑒𝑛𝑡)

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.



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

𝒚𝟏(𝒙, 𝒕) = 𝒂 𝐬𝐢𝐧(𝝎𝒕 − 𝒌𝒙)

The wave travelling along negative direction of x-axis can be represented as

𝒚𝟐(𝒙, 𝒕) = 𝐚 𝐬𝐢𝐧[𝛚𝐭 + 𝐤𝐱 + 𝛑] = −𝒂 𝒔𝒊𝒏(𝝎𝒕 + 𝒌𝒙)

The resultant wave on the string is, according to the principle of superposition:

𝑦(𝑥, 𝑡) = 𝑦1(𝑥, 𝑡) + 𝑦2(𝑥, 𝑡)

= 𝑎[𝑠𝑖𝑛(𝜔𝑡 − 𝑘𝑥) − sin( 𝜔𝑡 + 𝑘𝑥)]

Using the familiar trigonometric identity

𝐒𝐢𝐧(𝐀 − 𝐁) − 𝐒𝐢𝐧(𝐀 + 𝐁) = −𝟐𝐜𝐨𝐬𝐀𝐬𝐢𝐧𝐁

We get,

A = 𝜔𝑡 and B =kx

𝐲(𝐱, 𝐭) = −(𝟐𝐚 𝐬𝐢𝐧 𝐤𝐱) 𝐜𝐨𝐬 𝛚𝐭

Note: We can consider any of equations

𝒚(𝒙, 𝒕) = 𝒂 𝐬𝐢𝐧(𝝎𝒕 − 𝒌𝒙)

or

𝒚(𝒙, 𝒕) = 𝒂 𝐬𝐢𝐧(𝒌𝒙 − 𝝎𝒕)

There is a phase change of π i.e. the initial phase of y = 𝒂 𝐬𝐢𝐧(𝝎𝒕 − 𝒌𝒙) is π.

𝒚(𝒙, 𝒕) = 𝒂 𝐬𝐢𝐧(𝒌𝒙 − 𝝎𝒕)

𝒚(𝒙, 𝒕) = 𝒂 𝐬𝐢𝐧(𝝎𝒕 − 𝒌𝒙) = −𝒂 𝐬𝐢𝐧(𝒌𝒙 − 𝝎𝒕) = 𝒂 𝐬𝐢𝐧(𝒌𝒙 − 𝝎𝒕 + 𝝅)



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 𝟐𝒂 𝒔𝒊𝒏𝒌𝒙.

Thus in this wave pattern,

The amplitude varies from point to point, but each element of the string oscillates

with the same angular frequency ω 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.



From equation,

𝒚(𝒙, 𝒕) = −(𝟐𝒂 𝒔𝒊𝒏𝒌𝒙) 𝒄𝒐𝒔 𝝎𝒕

The positions of nodes (where the amplitude is zero) are given by

𝟐𝐚𝐬𝐢𝐧𝐤𝐱 = 𝟎

𝑠𝑖𝑛 𝑘𝑥 = 0

which implies

𝒌𝒙 = 𝒏𝝅; 𝒏 = 𝟎, 𝟏, 𝟐, 𝟑, ….

Since 𝒌 = 𝟐𝝅/𝝀,

we get

𝒙 =𝒏𝝀

𝟐; 𝒏 = 𝟎, 𝟏, 𝟐, 𝟑, ….

Clearly, the distance between any two successive nodes is 𝝀

𝟐.

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 𝑠𝑖𝑛 𝑘𝑥:

𝟐𝐚𝐬𝐢𝐧 𝐤𝐱 = 𝟏

which implies

𝒌𝒙 = (𝒏 + ½) 𝝅 ; 𝒏 = 𝟎, 𝟏, 𝟐, 𝟑, . ..

With 𝒌 = 𝟐𝝅/𝝀, we get

𝒙 = (𝒏 + ½)𝝀

𝟐 ; 𝒏 = 𝟎, 𝟏, 𝟐, 𝟑, … ..



Again the distance between any two consecutive antinodes is 𝝀

𝟐.

If the length of the string is 𝐿, then its first end can be taken as 𝑥 = 0 while the other end is denoted

as 𝑥 = 𝐿

Eq. 𝑥 =𝑛𝜆

2; 𝑛 = 0, 1, 2, 3, . .. can be applied to the case of a stretched string of length 𝐿 fixed at

both ends.

Taking one end to be at 𝑥 = 0, the boundary conditions are that 𝒙 = 𝟎 and 𝒙 = 𝑳 are positions

of nodes.

The 𝒙 = 𝟎 condition is already satisfied.

The 𝒙 = 𝑳 node condition requires that the length L is related to λ by

𝑳 = 𝒏 𝝀/𝟐 ; 𝒏 = 𝟏, 𝟐, 𝟑, . ..

Thus, the possible wavelengths of stationary waves are constrained by the relation

λ = 𝟐𝑳

𝒏; 𝒏 = 𝟏, 𝟐, 𝟑, …

with corresponding

frequencies can be obtained by using relation

𝒇 = 𝒗/ 𝝀

where 𝒗 is the speed of wave in the given medium.

𝒇 = 𝒏𝒗/𝟐𝑳, 𝒇𝒐𝒓 𝒏 = 𝟏, 𝟐, 𝟑,

We have thus obtained the natural frequencies - the normal modes of oscillation of the system.



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

𝐟𝟏 = 𝐯/𝟐𝐋

Here 𝑣 is the speed of wave determined by the properties of the medium.

The speed of a wave in a string 𝒗 = √𝐓

𝛍

T= tension,

μ = mass per unit length

Thus

𝐟𝟏 =𝟏

𝟐𝐋√

𝐓

𝛍

The n = 2 frequency is called the second harmonic or first overtone;

𝑓2 =2𝑣

2𝐿

𝑓2 = 2𝑓1

𝐧 = 𝟑 is the third harmonic or second overtone

𝐟𝟑 = 𝟑𝐯/𝟐𝐋

𝐟𝟑 = 𝟑 𝐟𝟏



Hence,

𝑓𝑛 = 𝑛 𝑓1; 𝑛 = 1,2,3 … … ..

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

https://youtu.be/Ew0fZh9INbQ



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 𝐻𝑧.

The linear mass density of the string is 4.0 × 10-2 kg / m and its mass is 2 × 10−2 𝑘𝑔.

Calculate:

( 𝑖 ) the velocity of a transverse wave on the string,

( 𝑖𝑖 ) the tension in the string.

SOLUTION:

Mass of the string, 𝑚 = 2 × 10−2 𝑘𝑔

Linear density of the string µ = 4 × 10−2 𝑘𝑔/𝑚

Frequency, 𝑓1 = 45 𝐻𝑧

Using Linear density of the string µ = mass / length

We know, length of the wire = (2 × 10−2)/(4 × 10−2) = 0.5 𝑚

as, 𝜆 = 2𝑙/𝑛

For fundamental node, 𝑛 = 1 => 𝜆 = 2𝑙 = 2 × 0.5 = 1𝑚

(i) Therefore, speed of the transverse wave,

Speed = frequency x wavelength

= 1 × 45 = 45 𝑚/𝑠

(ii) 𝑣 = √𝑇

𝜇

Tension in the string = µ𝑣2

= 4 × 10−2 × 452 = 𝟖𝟏 𝑵



EXAMPLE

A steel bar of length 200 𝑐𝑚 is nailed at its mid–point. The fundamental frequency of

longitudinal vibrations of the rod is 2.53 𝑘𝐻𝑧. At what speed will the sound be able to

travel through steel?

SOLUTION

Length, 𝑙 = 200𝑐𝑚 = 2𝑚

Fundamental frequency of vibration,

𝑓 = 2.53 𝑘𝐻𝑧 = 2.53 × 103𝐻𝑧

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 𝜆/2 => 𝑙 = 𝜆/2

or, 𝜆 = 2 × 2 = 4 𝑚

Thus, sound travels through steel at a speed of

𝑣 = 𝑓𝜆

𝑣 = 4 × 2.53 × 103 = 𝟏𝟎. 𝟏𝟐 𝒌𝒎/𝒔

EXAMPLE

The transverse displacement of a wire (clamped at both its ends) is described as:

y (x, t) = 0.06sin(2π3x)cos(120πt)

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 wire’s tension.

[x and y are in meters and t in secs]

SOLUTION: The general equation of a stationary wave is:



𝑦(𝑥, 𝑡) = 2𝑎 𝑠𝑖𝑛 𝑘𝑥 𝑐𝑜𝑠 𝜔𝑡

Comparing 𝑦(𝑥, 𝑡) = 0.06𝑠𝑖𝑛(2𝜋3𝑥)𝑐𝑜𝑠(120𝜋𝑡) with the general equation.

(i) The given function describes a stationary wave.

(ii) The transverse displacement of the wires is described as:

0.06𝑠𝑖𝑛(2𝜋3𝑥)𝑐𝑜𝑠(120𝜋𝑡)

Comparing with general, we get:

2𝜋/𝜆 = 2𝜋/3

wavelength 𝜆 = 𝟑𝒎

Also, 2𝜋𝑣/𝜆 = 120𝜋, speed 𝑣 = 𝟏𝟖𝟎 𝒎/𝒔

And, 𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 = 𝑣/𝜆 = 180/3 = 𝟔𝟎 𝑯𝒛

(iii) Velocity of the transverse wave, 𝑣 = 180𝑚/𝑠

The string’s mass, 𝑚 = 6 × 10−2 𝑘𝑔

String length, 𝑙 = 3 𝑚

Mass per unit length of the string, 𝜇 = 𝑚/𝑙 = (6 × 10−2)/3 = 2 × 10−2𝑘𝑔/𝑚

Let the tension in the wire be 𝑇

Therefore, 𝑇 = 𝑣2𝜇

= 1802 × 2 × 10−2

= 𝟔𝟒𝟖 𝑵

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.



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 Ɩ having mass per unit length m is stretched with a tension T, the

fundamental frequency of vibration 𝑓 is given by;

𝑓 =1

2𝐿√

𝑇

𝑚

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).

𝑓 ∝1

𝑙

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).

𝑓 ∝ √𝑇

Relation between frequency and length

From the law of length, f × Ɩ = constant

A graph between 𝑓 and 1/𝑙 will be a straight line.

Relation between length and tension

From the equation for frequency, √T / Ɩ = constant.

A graph between 𝑇 and 𝑙2 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

https://youtu.be/QXJ2oc4hx98



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



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

𝐟 =𝟏

𝟐𝐥√

𝐓

𝐦

Here, m = mass per unit length of the string;

𝑙 = 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,

𝒇 ∝𝟏

𝒍 𝒐𝒓 𝒇 𝒍 = 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕

(b) If frequency f is constant, for a given wire (m is constant), √𝑻

𝒍 is constant. That is 𝒍𝟐 ∝ T

Variation of resonant length with frequency of tuning fork

(i) VARIATION OF FREQUENCY WITH LENGTH

PROCEDURE



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 𝑛1 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 Ɩ

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/Ɩ where

Ɩ is the resonating length.



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 Ɩ for each fork and, calculate the reciprocals, 1

𝑙 of the resonating lengths Ɩ.

Plot 1

𝑙 vs n, taking n along x axis and

1

𝑙 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 Ɩ is found to be constant and the graph of 𝟏

𝒍 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



THINK ABOUT THESE

Error may occur in measurement of length Ɩ. 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 𝑙2 and T as shown

Graph between l2 and T

OBSERVATIONS (B)

Frequency of the tuning fork = ... Hz

Variation of resonant length with tension



CALCULATIONS AND GRAPH

Calculate the value of 𝑇/𝑙2 for the tension applied in each case. Alternatively, plot a graph of

𝑙2 vs T, taking 𝑙2 along y-axis and T along the x-axis.

RESULT

It is found that value of 𝑻/𝒍𝟐 is constant within experimental error.

The graph of 𝒍𝟐 vs T is found to be a straight line.

This shows that 𝑙2 α T or 𝑙 ∝ √𝑇.

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 Ɩ. 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.



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 Ɩ for each

of these for a given frequency, n and tension, T.

Plot a graph between the values of m and 𝟏

𝒍𝟐 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 𝑦(𝑥, 𝑡) = [2𝑎𝑠𝑖𝑛 𝑘𝑥 ] 𝑐𝑜𝑠 𝜔𝑡

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 λ/2.

A stretched string of length L fixed at both the ends vibrates with frequencies given

by 1

2

𝑣

2𝐿 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



the fundamental mode or the first harmonic. The second harmonic is the oscillation

mode with n = 2 and so on.

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Physics-02 (Keph 201504) - CIET