Chapter 15: waves

Introduction

If you drop a little pebble in a pond of still water, the water surface gets disturbed. The disturbance does not remain confined to one place, but propagates outward along a circle. If you continue dropping pebbles in the pond, you see circles rapidly moving outward from the point where the water surface is disturbed. It gives a feeling as if the water is moving outward from the point of disturbance. If you put some cork pieces on the disturbed surface, it is seen that the cork pieces move up and down but do not move away from the centre of disturbance.

This shows that the water mass does not flow outward with the circles, but rather a moving disturbance is created. These patterns, which move without the actual physical transfer or flow of matter as a whole, are called waves. 

For example: Consider a boy holding a thread and one end of thread is tied to the wall. When a boy moves the thread, the thread moves in the form of a wave.

In this Chapter, we will study such waves. In this chapter we will see the importance of waves in our life.

We will also study about the different properties of waves, some terms related to waves and also about different types of waves. We will also learn how waves propagate.

Waves

A wave is a kind of distribution due to repeated vibrations of particles from a normal or equilibrium condition, propagating without the transport of matter. In general a wave transports both energy and momentum. Waves transport energy and the pattern of distribution has information that propagates from one point to another.

For example:  Consider the sound of the horn; this sound reaches our ear because of sound waves. There is transfer of energy from one point to another with the help of particles in the medium.

• These particles don’t move, they just move around their mean position, but the energy is getting transferred from one particle to another and it keeps on transferring till it reaches the destination.
• The movement of a particle is initiated by the disturbance. And this disturbance is transferred from one point to another through space and time.

Note:-Energy and not the matter is transferred from one point to another.

Types of Waves

1. Mechanical waves
2. Electromagnetic waves
3. Matter waves

Mechanical waves: - The mechanical waves are governed by all of Newton’s laws of motion.  Medium is needed for propagation of the wave.   For Example: - Water Waves, Sound Waves

Electromagnetic waves: - Electromagnetic waves are related to electric and magnetic fields.  An electromagnetic wave does not need a medium to propagate, it carries no mass, does carry energy. Examples: - Satellite system, mobile phones, radio, music player, x-rays and microwave.

Matter waves: - Waves related to matter. Matter consists of small particles .Matter waves are associated with moving electrons, protons, neutrons & other fundamental particles etc. It is an abstract concept.

Types of wave motion

Depending on the relationship between the direction of oscillation of individual particles and wave propagation, the waves are classified in two categories. They are Transverse waves and longitudinal waves.

Transverse waves

The transverse waves are those in which the direction of disturbance or displacement in the medium is perpendicular to that of the propagation of the wave.

• The direction in which a wave propagates is perpendicular to the direction of disturbance.

For example: - Consider a man holding one end of a thread and the other end of the thread is fixed to the wall. When a little jerk is given to the thread in the upward direction. The entire thread moves in a wavy manner.

• The jerk propagated along the entire length of the thread.
• The small disturbance which came from the source at one end, that disturbance getting propagated and that is known as direction of propagation.

Disturbance is vertically upward and wave is horizontal. They are perpendicular to each other.  This type of wave is known as transverse wave.

Conclusion: Transverse waves are those waves which propagate perpendicular to the direction of the disturbance. Direction of disturbance is the direction of motion of particles of the medium.

Longitudinal waves

The waves in which individual particles of the medium executes simple harmonic motion about their mean positions along the directions of propagation of the wave are called longitudinal waves.

For example:  Waves in spring and sound waves etc.

Longitudinal means something related to length. In longitudinal waves the direction of disturbance or displacement in the medium is along the propagation of the wave.

• In a Longitudinal wave there are regions where particles are very close to each other. These regions are known as compressions.
• In some regions the particles are far apart. Those regions are known as rarefactions.

Displacement of progressive waves

Amplitude and phase together describe the complete displacement of the wave. Displacement function is periodic in space and time.

Displacement of the particles in a medium takes place along the y-axis. Generally displacement is denoted as a function of X and T, but here it is denoted by y.

In case of transverse wave displacement is given as: y(x, t)

Where x=propagation of the wave along x-axis, and particles oscillate along y-axis.

Therefore y(x, t) = A sin (kx – ωt + φ)

This is the expression for displacement. This expression is the same as the displacement equation which is used in oscillatory motion.

As cosine function; y(x, t) = B cos (kx – ωt + φ),

As both sine and cosine function) y (x, t) = A sin (kx – ωt + φ) + B cos (kx – ωt + φ).

Mathematically:

• Wave travelling along +X-axis:   y(x, t) = a sin (kx – ωt + φ).

As time t increases the value of x increases. This implies the x moves along the positive x-axis.

•   Waves travelling along -X axis: y(x, t) = a sin (kx +ωt + φ).

As time t decreases the value of x decreases. This implies the x moves along (-) ive x-axis.

Amplitude and Phase of a wave Amplitude and phase together describes the position of the particle.

Amplitude is the maximum displacement of the elements of the medium from their equilibrium positions as waves pass through them. It is denoted by A.

• In Transverse waves the distance between the point P and Q (in the Figure) is maximum displacement. This maximum displacement of the particles is known as amplitude.

In Longitudinal waves In case of longitudinal waves the particles will not oscillate to a very large distance. 

• Amplitude is the centre of two compressed regions. Because at the centre of the two compressed regions the particle is most free to displace to maximum displaced position.

Phase:  Phase of a wave describes the state of motion as the wave sweeps through an element at a particular position.

In-phase– Two points are said to be in-phase with each other when these two points are at the same position and they both are doing the same thing i.e. both the two points are exhibiting the same behaviour. Points C and F are in phase with each other.

Out-of-phase –Two points are said to be out of phase even though they are at the same points but they are doing opposite things i.e. both the points are exhibiting the different behaviour.

• Out of phase means which is not in phase.
• Points B and D,E and G are out of phase by 180 degree

Two waves can be completely in-phase or out of phase with each other. They can be partially in phase or out of phase with each other. Let’s try to understand the concept with help of an example

• Consider two points A and B on a wave. Their positions as well as their behaviour are the same. Therefore points A and B are in phase.
• Consider points A and C on a wave. They are not in phase with each other as their position is not the same.
• Similarly the points C and D are not in phase with each other as their positions are the same but the behaviour is different. Therefore they are not in phase with each other.
• Consider the points F and G. Their positions are the same but the behaviour is totally opposite. So F and G are out of phase.
• Consider the points F and H; they are in phase with each other as their position is the same as well as their behaviour.

Wavelength

Wavelength The term wavelength means length of the wave. Wavelength is defined as the minimum distance between two consecutive points in the same phase of wave motion. It is denoted by λ.

In case of transverse waves we use the term crest for the peak of the maximum displacement. The point of minimum displacement is known as trough.

In case of transverse wave wavelength is the distance between two consecutive crests or distance between two consecutive troughs.

In case of longitudinal waves, wavelength is the distance between the two compressions or the distance between the two rarefactions provided the compressions or rarefactions are nearest.

Wave Number

Wave number is the reciprocal of wavelength of a wave. It is defined as the number of waves per unit length. Since the unit of wavelength is a metre, the unit of wave number is the inverse of a metre.

Time Period, Frequency and Angular frequency

1. Time Period of a wave: - Time Period of a wave is the time taken through one complete oscillation. It is denoted by ’T’.
2. Frequency of a wave: - Frequency of a wave is defined as the number of oscillations per unit time. It is denoted by f.   f =1/T.
3. Angular frequency: -Angular frequency is defined as the frequency of the wave in terms of a circular motion. The term angular frequency is used only when there is an angle involved in the motion in that particular motion .It is denoted by ‘ω’.

Relation between  w , T and f  is given by   w=2πf =2π/T

 

Travelling Waves Travelling waves are waves which travel from one medium to another.  They are also known as progressive waves. Because they progress from one point to another.

• Both longitudinal and transverse waves can be travelling waves.
• Wave as a whole moves along one direction.

Speed of a transverse wave in a stretched string

Consider a stretched string and if given transverse disturbance on one end then the disturbance travels throughout the string. Thereby giving rise to transverse waves. The particles move up and down and the waves travel perpendicular to the oscillation of the particles.

Transverse wave speed determined by two factor which are:

1. Mass per unit length- As mass gives rise to Kinetic energy. If no mass then no kinetic energy. Then there will be no velocity.  It is denoted by μ.
2. Tension-Tension is the key factor which makes the disturbance propagate along the string. Because of tension the disturbance travels throughout the wave. It is denoted by T.

Dimensional Analysis to show how the speed is related to mass per unit length and Tension

μ = [M]/ [L] …… (i)

T=F=ma = [MLT^-2] ….(ii)

Dividing equation (i) by (ii):-

Therefore   , where C=dimensionless constant

Conclusion: velocity v depends on properties of the medium and not on frequency of the wave.

Speed of a longitudinal wave in a stretched string

Longitudinal wave speed determined by:

Density – Longitudinal waves are formed due to compressions (particles very close to each other) and rarefactions (particles are far from each other).

At certain places it is very dense and at certain places it is very less dense. So density plays a very important role. It is denoted by ρ.

Bulk modulus– Bulk modulus tells how the volume of a medium changes when the pressure on it changes.

If we change the pressure of compressions or rarefaction then the volume of the medium changes. It is denoted by B.

Dimensional Analysis to show how the speed is related to density and bulk modulus

ρ =mass/volume= [ML^-3]

B = - (Change in pressure (ΔP))/Change in volume (ΔV/V))

ΔV/V is a dimensionless quantity as they are 2 similar quantities.

ΔP= F/A=[MLT^-2]/[L²]=[ML^-1T^-2]

Dividing ρ/B= [ML^-3]/[ML^-1T^-2]=[L^-2T²]=[L^-1T]²

We know that v=[LT^-1]   so   ρ/B α  1/v²

v² α   B/ρ  So 

Where C=dimensionless constant

In case of fluids: - v= C √ (B/ ρ)

In case of solids: - v= C √ (Y/ ρ)

The principle of superposition of waves

Principle of superposition of waves describes how the individual waveforms can be algebraically added to determine the net waveform.

Waveform tells about the overall motion of the wave. It does not tell about individual particles of the wave.

Suppose we have 2 waves and Example of superposition of waves is Reflection of waves.​​​​​​​

Mathematically: -

Case1:- Consider 2 waves which are in phase with each other. They have the same amplitude, same angular frequency, and same angular wave number.

If wave 1 is represented by y1(x, t) =a sin (kx – ωt).

Wave 2 is also represented by y2(x, t) =a sin (kx – ωt).

By the principle of superposition

The resultant wave   y(x t) = (2a sin (kx – ωt)) will also be in phase with both the individual waves but the amplitude of the resultant wave will be more.

Case2:- Consider when the two waves are completely out of phase. φ = π

If wave 1 is represented by y1(x, t) =a sin (kx – ωt).

Wave 2 is represented by y2(x, t) =a sin (kx – ωt+ π).

Y2 =a sin (π-(-kx+ ωt) which gives y2=-a sin (kx- ωt)

Therefore by superposition principle y=y1+y2=0

Case 3:- Consider when the two waves partially out of phase φ>0; φ<π

If wave 1 is represented by y1(x, t) =a sin (kx – ωt).

Wave 2 is represented by y2(x, t) =a sin (kx – ωt+ φ).

Therefore by the principle of superposition of waves,

y = y1+ y2   = a [sin (kx-ωt) +sin (kx – ωt+ φ]

y=2a cos (φ/2) sin (kx – ωt + (φ/2))

(By using the formula sinA +sinB=2sin (A+B)/2) cos (A-B)/2)

Amplitude = 2a cos (φ/2) and Phase will be determined by (φ/2).

Reflection of waves

Reflection of waves is the change in the direction of a wave upon striking the interface between two materials. When a wave strikes any interface between any two mediums the bouncing back of the wave is termed a reflection of waves.

The interface can be categorized into 2 types:

Open boundary: - When a wave strikes an interface in the case of open boundary it will get reflected as well as refracted.

Closed boundary or a rigid boundary: - When a wave is an incident on an interface it will completely get reflected. Example:-Wave striking wall (echo)

Reflection at the rigid boundary

Consider a string which is fixed to the wall at one end. When an incident wave hits a wall, it will exert a force on the wall.

By Newton’s third law, the wall exerts an equal and opposite force of equal magnitude on the string.

Since the wall is rigid, the wall won’t move, therefore no wave is generated at the boundary. This implies the amplitude at the boundary is 0.

As both the reflected wave and incident wave are completely out of phase at the boundary. Therefore φ=π.

Therefore, the incident wave is yi(x, t) = a sin (kx – ωt),

Reflected wave is yr(x, t) = a sin (kx + ωt + π) = – a sin (kx + ωt)

By superposition principle y= yi + yr =0

Conclusion: -

The reflection at the rigid body will take place with a phase reversal of π or 180.

Reflection at the open boundary

The reflection at an open boundary will take place without any phase change.  In this case, a boundary pulse is generated. Therefore amplitude at the boundary is maximum.

• This means the reflected wave and incident wave are in phase with each other. As a result the phase difference φ=0.
• Therefore, the incident wave is  yi(x, t) = a sin (kx – ωt),
• Reflected wave is yr(x, t) = a sin (kx – ωt).
• By superposition principle y= yi + yr =2a sin (kx – ωt)

Standing wave

A stationary wave is a wave which is not moving, i.e. it is at rest.

When two waves with the same frequency, wavelength and amplitude traveling in opposite directions will interfere they produce a standing wave.

Conditions to have a standing wave: - Two traveling waves can produce a standing wave if the waves are moving in opposite directions and they have the same amplitude and frequency.

• At certain instances when the peaks of both the waves will overlap. Then both the peaks will add up to form the resultant wave.
• At certain instances when the peak of one wave combines with the negative of the second wave . Then the net amplitude will become 0.
• As a result, a standing wave is produced. In the case of a stationary wave, the waveform does not move.

Mathematically,

• Wave travelling towards left yl(x,t) =a sin(kx– ωt) and towards right yr(x,t) =a sin (kx + ωt)

The principle of superposition gives, for the combined wave

y (x, t) = yl(x, t) + yr(x, t) = a sin (kx – ωt) + a sin (kx + ωt)

Y(x, t) = (2a sin kx) cos ωt (By calculating and simplifying)

The above equation represents the standing wave expression.

Amplitude = 2a sin kx.

• The amplitude is dependent on the position of the particle.
• The cos ωt represents the time-dependent variation or the phase of the standing wave.

Below is the difference between traveling/progressing waves and standing waves

Nodes and Antinodes of Standing Wave

The amplitude of a standing wave doesn’t remain the same throughout the wave. It keeps on changing as it is a function of x.

At certain positions the value of the amplitude is maximum and at certain positions, the value of the amplitude is 0.

• Nodes: - Nodes represent the positions of zero amplitude.
• Antinodes: - Antinodes represent the positions of maximum amplitude.

Characteristics of standing wave:

• There are certain points in the medium in a standing wave, which are permanently at rest; these are called nodes.  The distance between two consecutive nodes is λ/2.
• There are certain points in standing waves that have maximum amplitude. These are called antinodes. The distance between two consecutive antinodes is also λ/2.
• The wavelength and the time period of the stationary waves is the same as that of the component waves by which it is formed.

Nodes and Antinodes: system closed at both ends

System closed at both ends means both ends are rigid boundaries.

Whenever there is a rigid body there is no displacement at the boundary. This implies the boundary amplitude is always 0. Nodes are formed at the boundary.

Standing waves on a string of length L fixed at both ends have restricted wavelength. This means the wave will vibrate for certain specific values of wavelength.

At both ends, nodes will be formed, so Amplitude=0.

Expression for node x = (nλ)/2.This value is true when x is 0 and L.

When x=L:- L=(nλ)/2 =>λ=(2L)/n ; n=1,2,3,4,…..

λ cannot take any value but it can take values that satisfy λ= (2L)/n this expression.

That is why we can say that the standing wave on a string which is tied on both ends has a restricted wavelength. As the wavelength is restricted therefore wavenumber is also restricted.

Here n=1 is a fundamental mode of vibration (first harmonics)

Similarly n= 2, 3, and 4 are called second, third and fourth harmonics respectively and so on.

Frequency =velocity / wavelength f= v/λ    

Corresponding frequencies which a standing wave can have is given as:   

 

Where v= speed of the traveling wave. These frequencies are known as natural frequencies or modes of oscillations.

Vibrations of Air Column

The vibrating air column in organ pipes is a common example of stationary waves. An organ pipe is a cylindrical tube which may be closed (at one end) or open at both ends (open organ pipe).

If the air in the pipe at its open end is made to vibrate longitudinal waves are produced. These waves travel along the pipe towards its far end and are reflected back. Thus due to the superposition of incident and reflected waves, stationary waves are formed in the pipe.

Closed organ pipe

If one end of the pipe is closed, the reflected wave is 180 degrees out of phase with the incoming wave. This displacement of the small volume elements at the closed end must always be zero. Hence the closed end must be a displacement node.

Open organ pipe

If both ends of the pipes are open and the system of air is directed against an edge, standing longitudinal waves can be set up in the tube. The open end has displacement antinodes

Various modes of vibration of air column in an open organ pipe are shown below

Beats

This phenomenon arises from the interference of waves having nearly the same frequencies.

The periodic variation on the intensity of the sound wave caused by the superposition of two sound waves of nearly the same frequencies and amplitude traveling in the same direction are called beats.

One rise and one fall in the intensity of sound constitute one beat and the number of beats per second is called beat frequency.

The frequency of two sources or two waves should not differ by more than 10 Hz, because if it is more than rising and fall in intensity of sound due to persistence of hearing.

If f1 and f2 are the frequencies of the two waves such that (f1 > f2) the

Beat frequency f _beat = f1 - f2

Doppler’s Effect

Doppler Effect is the phenomenon of motion-related frequency change.

Consider if a truck is coming from a very far-off location as it approaches our house, the sound increases and when it passes our house the sound will be maximum. And when it goes away from our house the sound decreases. This effect is known as the Doppler Effect.

In other words, the apparent change in frequency heard by the observer due to relative motion between source and the observer is known as the Doppler Effect.

A person who is observing is known as Observer and the object from which the sound wave is getting generated is known as Source.

When the observer and source come nearer to each other as result waves get compressed. Therefore wavelength decreases and frequency increases.

Doppler Effect will be analyzed under three cases.

Case 1:- Observer is stationary but the source is moving.

Case 2:-Observer is moving but the source is stationary.

Case 3:- Both the observer and the source are moving.