Waves & vibrations

ByKavita Monga

( Lecturer Physics)Govt. Polytechnic College,

Mohali (Khunimajra)Date: 04/10/12

Apr 19, 2023

WavesConcept of vibrations extends into the phenomenon of wave motion.

Water waves

Light waves

Sound

Radio

String

What Is a Wave?

• A wave may be thought of as a spreading of disturbance from one place to another.

• “Ripples.”

• As ripples spread out, they carry along with them, energy.

Waves are everywhere. We encounter waves on a daily basis. Everyday we come across sound waves, visible light waves, radio-waves, micro waves, water waves, waves in a string which are some of the examples. Also, there are certain phenomenon that resemble with waves. For example, the motion of pendulum of a clock, the motion of a mass suspended by a spring, a child taking swings etc. Our understanding of the physical world is incomplete until and unless we understand the nature, properties and behaviour of waves.

Wave motion is a recurring disturbance that advances through space with or without the use of a physical medium. So, wave motion is a means of transferring energy from one point to another point. For example, when a microphone is struck by sound waves, it converts sound energy into electrical energy. Similarly, when radio waves come across an antenna, they get converted to electrical energy. This means that sound, light, radio waves, waves in ocean or water bed when struck by a stone, waves in a rope, are all forms of energy that are moved by wave motion.

The most exciting example of wave motion is the “wave” that spectators create during a cricket match or any sport events, by standing up and sitting down, while chearing up their team, at certain regular intervals. The successive action of people standing up and sitting down create both the energy and the wave. If the stadium was empty i.e. nobody is there, no wave will exist and no energy can be transmitted. So, the people are the medium. Now, each person just stands up and sits back in his own seat. This means that the person i.e. medium does not leave its place whereas the wave travels forward.

General definitions of vibrations and waves

• Vibration: in a general sense, anything that switches back and forth, to and fro, side to side, in and out, off and on, loud and soft, or up and down is vibrating. A vibration is a wiggle in time.

• Wave: a wiggle in both space and time is a wave. A wave extends from one place to another.

• Vibrations and waves: the source of all waves is something that is vibrating. Waves are propagations of vibrations throughout space.

Wave Motion• medium - the stuff that carries the wave

Waves Medium

water waves waterwaves on a rope ropestadium waves people

sound airlight space (vacuum)

Wave Motion

Wave Motion

A wave travels along its medium, but the individual particles just move up and down.

Wave Motion (defn.)

Wave motion is a form of disturbance which travels through the medium due to the repeated periodic motion of the particles of the medium about their mean positions, the disturbance being handed over from one particle to the next adjoining particle and so on, without any net transport of the medium.

In a wave motion, the physical condition due

to a disturbance generated at some point in the medium is propagated to other points in the medium. The wave may be of any type, irrespective of that, the particles of the medium vibrate about their mean positions. So, in the case of wave motion, it is not matter that is propagated but it is only the state of motion of the matter that is propagated. In other words, we can say that it is a form of dynamic condition that is propagated from one point to the other point in the medium. So, in a wave motion, momentum and energy are propagated.

Wave Motion (defn.)

Characteristics of wave motion

1. Wave motion is a disturbance produced in the medium by the repeated periodic motion of the particles of the medium.2. There is a regular phase change between the various particles of the medium. Each particle starts vibrating a little later than the particle just ahead of it.3. The velocity of the wave is different from the velocity with whichthe particles of the medium are vibrating.

Contd.

4. Only the wave travels forward, whereas

the particles of the medium vibrate

about their mean positions.

5. In wave motion, momentum and energy

are transferred or propagated.

6. The medium through which the wave

propagates should possess the

properties of elasticity, inertia and

negligible frictional resistance.

Characteristics of wave motion

Types of WavesTwo types :(i) Elastic waves or mechanical waves. These are those waves which can be produced or propagated only in a material medium. Waves on water surface, waves on strings, sound waves etc. are all mechanical waves.(ii) Electromagnetic waves or non-mechanical waves. These are those waves which can pass even through vacuum i.e., these waves do not require any material medium to travel through. For example, radio waves, X-rays, microwaves, gamma rays etc.

The mechanical waves are of two types :

(i) Transverse waves and

(ii) Longitudinal waves

A transverse wave motion is that type of wave motion in which the particles of the medium vibrate at right angles to the direction of propagation of the wave.

Types of Waves

In longitudinal wave motion the particles of the medium vibrates in the same direction as the wave is moving.

Types of Waves: Transverse and Longitudinal

The motion of particles in a wave can either be perpendicular to the wave direction (transverse) or parallel to it (longitudinal).

Transverse Waves

Transverse Wave – The motion of the particles is at a right angle (perpendicular) to the

direction the wave is moving.

Example:

water waves

Transverse Wave Animation

In a transverse wave, the vibration of the individual particles of the medium is perpendicular to the direction of wave propagation.

In a transverse wave, the vibration of the individual particles of the medium is perpendicular to the direction of wave propagation.

Motion of particles

Motion of wave

Transverse Waves

• Travel in a direction perpendicular to the direction of vibration.

• Displacement of the particles is at right angles to the direction of travel of wave motion.

• Example: Transverse waves created by vertical swinging of a rope fixed at one end.

Transverse Waves

• Made up of crests and troughs.

• Example: rope waves, water waves, light waves and radio waves, infra red waves, etc.

In transverse wave motion crests and

troughs are formed.

• A crest is a portion of the medium temporarily raised above the mean position and its centre is the position of maximum displacement in the positive direction. Points A and C in the figure are centers of crests.

• A trough is a portion of the medium depressed temporarily and the centre of the trough is the position of maximum displacement in the negative direction. The points B and D in the figure are centers of successive troughs.

Examples of transverse waves

(i) When a tabla is struck, the movement of

its membrane produces transverse

waves.

(ii) When a rope held at one end is slightly

vibrated, the waves produced are

transverse.

(iii) When string of a Guitar or Sitar is struck

by the finger, the movement of string

produces these waves.

Characteristics of transverse waves1. The particles of the medium vibrate at

right angles to the propagation of the wave.

2. There is a regular phase difference between the particles of the medium.

3. The waves travel in the form of crests and troughs.

4. These waves are possible in a medium which possesses elasticity of shape

i.e. in solids and liquids only.

5. They transfer energy.

Wave TypesLongitudinal Wave – The motion of the particles are in the same direction as the wave is moving. The wave pulsates back and forth in the direction it is moving.

Example:

sound wavesLongitudinal Wave Animation

In a In a longitudinal wave,longitudinal wave, the vibration of the the vibration of the individual particles is parallel to the direction of individual particles is parallel to the direction of wave propagation.wave propagation.

In a In a longitudinal wave,longitudinal wave, the vibration of the the vibration of the individual particles is parallel to the direction of individual particles is parallel to the direction of wave propagation.wave propagation.

Motion of particles

Motion of wave

v

Longitudinal Waves

• Travel in a direction parallel to the direction of vibration.

• Displacement of the particles is in line with or parallel to the direction of travel of wave motion.

• Example: Longitudinal waves created by horizontal motion of a spring fixed at one end.

Longitudinal Waves

• Made up of compressions and rarefactions.

• Example: sound waves.

Example of what a speaker does to air molecules to produce sound

Longitudinal wave motion

• A longitudinal wave motion is that type of wave motion in which the particles of the medium vibrate along the direction of propagation of the wave.

• In longitudinal wave motion compressions and rarefactions are formed.

• A compression is a region of the medium in which particles come closer and the density of the medium around these particles is greater than the normal density. So, this region is formed when particles of the medium move in the same direction as the propagation of the wave. The regions represented as “C” in fig. are compressions.

• A rarefaction is a region in which the distance between the particles of the medium is more i.e. they get farther apart and the density of the medium around these particles is less than the normal density. So, this region is formed when particles move in opposite direction to the propagation of the wave. The regions represented as “R” in fig. are rarefactions.

Characteristics of Longitudinal waves

1. The particles of the medium vibrate simple harmonically along the same direction as the propagation of the wave.

2. There is a regular phase change between the particles of the medium.3. The waves travel in the form of compressions and rarefactions.4. These are possible in a medium

possessing elasticity of volume i.e. in solids, liquids and gases.

5. They transfer energy.

Wave MotionWave characteristics:• Amplitude, A• Wavelength, λ• Frequency f and period T• Wave velocity

Properties of Wave Motion • Crests (high points) and troughs (low

points) of transverse waves• Amplitude A: maximum displacement from

rest position in either direction. SI unit: metres(m)

Properties of Wave Motion

• Phase: Two points in phase move in the same direction with same speed and have same displacement from rest position (See red dot).

Properties of Wave Motion

• Wavelength: shortest distance between any two point on a wave that is in phase. SI unit: metres (m)

Properties of Wave Motion• Frequency, f: number of complete waves produced per

second. SI unit: hertz (Hz)• Period, T: time taken to produce one complete wave.

SI unit: second (s)

T = 1/f

T

Wave Equation

• Wave speed, v: distance travelled by a wave in one second. SI unit: m/s.

Speed = distance / time. V = / T.

But f = 1 / T.

v = f

Important definitions concerning wave motion

Wavelength (λ)

Wavelength of a wave is the length of one wave. It is the distance travelled by the wave during the time, any one particle of the medium completes one vibration about its mean position. We may also define wavelength as the distance between any two nearest particles of the medium, vibrating in the same phase.

In transverse wave motion, (λ) = distance between the centers of two consecutive crests or distance between the centers of two consecutive troughs. We can also define wavelength as the distance in which one crest and one trough are contained.

In longitudinal wave motion, (λ) = distance between the centers of two consecutive compressions or distance between the centers of two consecutive rarefactions. Also, wavelength can be defined as the distance in which one compression and one rarefaction are contained.

Frequency (ν)

Frequency of vibration of a particle is defined as the number of vibrations completed by a particle in one second. Now, one vibration is equal to one wavelength and so we can define frequency as the number of complete wavelengths completed by the wave in one second.

Time period (T)

It is the time taken by the particle to complete one vibration about its mean position. Now, as already started, one vibration is equivalent to one wavelength. So time period of a wave is the time taken by the wave to travel a distance equal to one wavelength.

Amplitude (a) It is the maximum displacement of the particle from its mean position.

Phase The phase of a vibrating particle at any instant is its state or condition regarding its

position and direction of motion at that instant.

It is measured as the ratio of the displacement of the particle at any instant to the amplitude of the particle or it is the fraction of the time interval that has elapsed since the particle crossed the mean position of rest in the positive direction or it is also equal to the angle swept by the radius vector since the vibrating particle last crossed its mean position of rest.

Phase of particle P is

4

T

8or

Phase difference The difference in the phases of two vibrating particles is called phase difference. If the two particles are in the same positions and move in the same directions, they are said to be in the same phase and if their positions or directions are different, they are said to be out of phase i.e. there is a phase difference.

• For example, particle at y has a phase of π/2 or T/4 and particle at X’ has a phase of π or T/2.

• Then phase difference (Φ) between them is = π– π/2 = π/2 or Φ = T/2 – T/4 = T/4

Relationship between frequency and time period

Time to complete n vibrations = 1 sec.

Time to complete one vibration sec.

From definition of time period, time taken to complete one vibration is called the time period (T).

T or ν

1

1

1

T

Relationship between velocity, frequency and wavelength

Velocity of the wave is the distance travelled by the wave in one second.

Velocity

Wavelength (λ) is the distance traveled by the wave in one time period (T).

Velocity v

distance

time

wavelengthtime period

ν

v

But = n

v = nλ

T

×1

T1

T

Sound Waves• Sound is a vibratory motion of the air. In our

ears, it gives the sensation of hearing. It is excited by vibrating bodies and transmitted to our ears without sensible motions of the air.

• The source is some object that causes a vibration.

• The vibration disturbs the particles in the surrounding medium, these particles then disturb those next to them, and so on.

• The wave carries the sound energy through the medium. Because sound is the vibration of matter, it does not travel through a vacuum or in outer space.

• The back and forth movement of a loudspeaker cone, guitar string or drum membrane results in compression waves of sound. When we speak our vocal cords vibrate creating sound.

• Sound can also be created by vibrating an object in a liquid such as water or in a solid such as iron. A train rolling on a track also creates a sound wave that travels through the tracks. They will then vibrate, creating sound in air that we hear even when the train is at a far off distance.

• Sound and music are parts of our everyday sensory experience. Just as humans have eyes for the detection of light and color, so we are equipped with ears for the detection of sound. The basis for an understanding of sound, music and hearing is the physics of waves. Sound is a wave which is created by vibrating objects and propagated through a medium from one location to another.

• The vibrating object which creates the disturbance could be the vocal chords of a person, the vibrating string and sound board of a guitar or violin, the vibrating tines of a tuning fork, or the vibrating diaphragm of a radio speaker.

• A tuning fork is a metal object consisting of two prongs capable of vibrating when struck by a rubber hammer. As the prongs of the tuning fork vibrate back and forth, they begin to disturb surrounding air molecules. These disturbances are passed on to adjacent air molecules by the mechanism of particle interaction.

Light Waves

• Light is everywhere in our world. It carries information from the things we see, to our eyes and finally to our brain. It seems to be so simple a phenomenon but when we sit to study it, it is a much more perplexing phenomenon.

• Newton believed that light is a collection of particles. He believed that light emits tiny corpuscles which travel in straight lines. The strongest evidence of particle nature of light was that light seems to travel in straight lines and cast shadows.

• The Dutch physicist Christian Huygens, suggested that light is a wave phenomenon. Huygens' proposal remained in a dump for almost about a century because the scientific community had a great faith in Newton's writings. Huygens' was later supported by double slit experiment of Thomas Young and experiments on diffraction of light by French physicist Augstin Jean Fresnel.

Light exhibits certain behaviors which are characteristic of any wave. Light reflects in the same manner that any wave would reflect. Light refracts in the same manner that any wave would refract. Light diffracts in the same manner that any wave would diffract. Light undergoes interference in the same manner that any wave would interfere. And light exhibits the Doppler effect just as any wave would exhibit the Doppler effect. Light behaves in a way that is consistent with our conceptual and mathematical understanding of waves.

But the wave theory of light failed to explain the phenomenon such as photoelectric effect, Compton effect, etc. Albert Einstein, in 1905 explained the photoelectric effect on particle nature of light. He considered that light consists of particles having no mass but carrying energy as well as momentum and were called photons / quanta. This gave rise to another concept i.e. Quantum nature of light.

Electromagnetic waves

Light waves are called electromagnetic waves. Electromagnetic waves are waves which are capable of travelling through a vacuum. Unlike mechanical waves which require a medium in order to transport their energy, electromagnetic waves are capable of transporting energy through the vacuum of outer space.

Electromagnetic waves are produced by a vibrating electric charge and as such, they consist of both an electric and a magnetic component. Light waves are fluctuations of electric and magnetic fields in space as shown

Electromagnetic spectrum

The electromagnetic spectrum is made up of six different types of waves. Radio waves, Microwaves, Infrared waves, Visible light, Ultraviolet light, X-rays and Gamma rays.

Visible Light

• Visible light waves are the only electromagnetic waves we can see. We see these waves as the colors of the rainbow. Each color has a different wavelength. Red has the longest wavelength and violet has the shortest wavelength. When all the waves are seen together, they make white light.

When white light shines through a prism, the white light is broken apart into the colors of the visible light spectrum. Water vapour in the atmosphere can also break apart wavelengths creating a rainbow.

Each color in a rainbow corresponds to a different wavelength of electromagnetic spectrum.

Ultraviolet Waves

Ultraviolet (UV) light has shorter wavelengths than visible light. Though these waves are invisible to the human eye, some insects can see them.

Radio Waves

Radio waves have the longest wavelengths in the electromagnetic spectrum. Radio waves are responsible for carrying signals to your radio, television and cellular phones.

Microwaves

• Microwaves have wavelengths that can be measured in centimeters. The longer microwaves, those closer to a foot in length, are the waves which heat our food in a microwave oven.

• Microwaves are good for transmitting information from one place to another because microwave energy can penetrate haze, light rain and snow, clouds, and smoke.

• Shorter microwaves are used in remote sensing. Microwaves, used for radar, are just a few inches long.

X-rays

As the wavelengths of light decrease, they increase in energy. X-rays have smaller wavelengths and therefore higher energy than ultraviolet waves. X-ray light tends to act more like a particle than a wave. X-ray detectors collect actual photons of X-ray light - which is very different from the radio telescopes that have large dishes designed to focus radio waves. X-rays are used to locate bone fractures as they can pass through flash but not bones. To study the structure of a crystal.

Gamma-rays

Gamma-rays have the smallest wavelengths and are the more energetic. These waves are generated by radioactive atoms and in nuclear explosions. Gamma-rays can kill living cells, a fact which medicine uses to its advantage, using gamma-rays to kill cancerous cells.

Periodic Motion• Periodic motion of a body is that motion

which repeats itself over and over again after a fixed interval of time. The fixed or regular interval of time after which the periodic motion is repeated again, is called its time period.

Examples of periodic motion • The bob of a pendulum, when displaced from

its mean position.

• A freely suspended bar magnet, when displaced from its equilibrium position along NS-line, repeats again and again.

• The rotation of earth about its axis with period of rotation of one day is a periodic motion.

•The motion of moon around the earth is a

periodic motion with a time period

= 27·3 days.

•The revolution of earth around the sun with a

period of one year.

• The phases of moon represent periodic motion.

Oscillatory Motion

• A motion that repeats itself over and over again about its mean position, such that it remains confined within two well defined limits called extreme positions on either side of the mean position (a fixed point), in a definite interval of time.

• A periodic and bounded motion of a body about its mean position is called an oscillatory or vibratory motion. So, it follows that all oscillatory motions are periodic motions but all periodic motions are not oscillatory.

Examples of Oscillatory Motion• When a liquid in a U-tube is displaced, it

executes oscillatory motion.• When a load attached to a spring is pulled

once a little from its fixed point and left, it executes oscillatory motion.

• The motion of pendulum of a wall clock.• If a piece of wood floating on the surface of

liquid is pressed down and released, it executes oscillatory motion.

• When a glass ball is dropped along walls of a semi-hemispherical bowl and released, it executes oscillatory motion.

Simple Harmonic Motion

If the acceleration of a body be proportional to its displacement from its mean position or any other fixed point in its path and be always directed towards it, the body is said to execute a simple harmonic motion.

Mathematically, a linear SHM may be regarded as the projection of a uniform circular motion on the diameter of the circle, or any other fixed line in the plane of the circle. This circle being referred to as the circle of reference and may be purely imaginary.

Examples of simple harmonic motion

Linear Simple Harmonic Motion

(i) The up and down oscillations of the piston of a cylinder, containing a gas, when suddenly pressed down and released.

(ii) The oscillations of mercury or water contained in a U-tube, when the column in one limb is depressed and released.

(iii) The vertical oscillations of an elastic string or a spiral spring suspended from a rigid support and loaded at its lower end.

Angular Simple Harmonic Motion

(i) The oscillations of a pendulum, provided the amplitude is small,

(ii) The oscillations of a magnet suspended in a magnetc field,

(iii) Torsional oscillations.

Examples of simple harmonic motion

Displacement

in

SHM

Y = r sin θ

Y = r sin wt