### Wave specifications

- Definition: “A wave is a disturbance that travels in a medium (e.g. air, water etc.)”
- Source: A wave is initiated by a vibrating object and travels away from it.
- Particles of the medium: vibrate about their rest position at the same frequency as the source.
- A wave transfers energy and momentum, but never mass.
- Medium: No large scale movement of the medium as the wave passes through it.

### Wave properties

- Wavelength (λ): Shortest distance between two points that are in phase on a wave.
- Two consecutive crests or two consecutive troughs.

- Frequency (f): Number of vibrations per second performed by the source of waves.
- Period (T): Time taken for one complete wavelength to pass a fixed point. T = 1/f.
- Displacement (x): Instantaneous distance of the moving object from its mean position (in a specified direction).
- Amplitude (A/xo): Maximum displacement of wave from its rest position.
- Speed (v/c): Depends only on the properties of the medium and not the source.
- v = λf or c (speed of light) = λf.

### Graphs:

**Displacement-distance:**

**Displacement-time:**

### Wave classification

- Mechanical waves: Require a medium to travel through.
- Sound: Constant velocity (sqrt(v) proportional to temperature), longitudinal.
- Hearable: 20 Hz to 20000 Hz.
- Frequency: Pitch.
- Amplitude: Volume (Loudness).

- Sound: Constant velocity (sqrt(v) proportional to temperature), longitudinal.
- Electromagnetic waves: May travel in vacuum.
- Speed of light (c): 3 x 10^8 ms^-1.
- Wavelength: Colour.
- Amplitude: Brightness.

### Electromagnetic spectrum:

- Transverse waves: Displacement of particles is perpendicular to the direction of energy transfer. Both electromagnetic and mechanical waves (e.g in a rope).
- Longitudinal waves: Displacement of particles is parallel to the direction of energy transfer. Only mechanical waves, made of compression and rarefaction.

### 4.3 Wave Characteristics

Wavefronts: “Surfaces/lines that join points with the same phase.”

Rays: “Lines in the direction of energy transfer.”

Wavefronts and rays are perpendicular to each other.

Intensity (I)

- Definition: “The intensity of a wave at a point P is the amount of energy arriving at P per unit area per unit time.”
- Equation: I = Power/Area = P/4πr^2, where r is the distance.
- Units: J s^-1 m^-2 or W m^-2.
- Inverse-square law: Doubling the distance reduces power received by a quarter.
- Intensity is proportional to amplitude squared.

### Superposition

- Definition: “When two or more waves collide, the total displacement is the vector sum of their individual displacements”.

Reflection of pulses

- Fixed end: Pulse inverts, due to reaction force (e.g. of the wall).

Free end: Pulse does not invert.

### Polarization

- Definition: “An electromagnetic wave is said to be plane polarized if the electric field oscillates on the same plane”.
- Occurrence: It only occurs in transverse waves, e.g. light, which is normally unpolarized.

- Polarization of unpolarized wave: Original intensity reduced by half.

- Methods of polarization:
- Passing through a polarizer and an analyzer.
- Polarizer: Device that produces plane-polarized light from an unpolarized beam.
- Analyzer: Polarizer used to detect polarized light.

- Reflection on non-metallic surfaces (e.g. a lake): Partial polarization into different components, the greatest in the plane parallel to the non-metallic surface.

- Passing through a polarizer and an analyzer.
- Optically active substance (e.g. sugar solutions): Rotates the plane of polarization, normally placed between the polarizer and analyzer.
- Sugar solution: Length and concentration of solution is proportional to the angle of rotation.
- Liquid-crystal displays (LCDs): liquid crystal is sandwiched between two glass electrodes. Rotates the plane of polarization according to pd across it.

- Polarimeter: Measures the intensity after the analyzer.
- Malus’ Law (for already polarized light): I = Io cosθ^2 and E = Eo cosθ^2, where θ is the angle between the incident wave and the polarizer or analyzer.

- Brewster’s Law: If the reflected ray and the refracted are at right angles to one another, then the reflected ray is totally polarized. Read about the reflection and refraction of waves.
- The angle of incidence for this condition is known as the polarizing angle.
- θi + θr = 90º and n = sinθi/sinθr = sinθi/cosθi = tanθi.

- Uses of polarization:
- Polaroid sunglasses: Allows waves with a vertical plane of polarization and absorbs waves with an horizontal plane of polarization.
- Reduces glare from non-metallic surfaces.

- Stress analysis: When white light is passed through stressed plastics, colored lines are observed in regions of maximum stress.

- Polaroid sunglasses: Allows waves with a vertical plane of polarization and absorbs waves with an horizontal plane of polarization.

### 4.4 Wave Behavior

Reflection and refraction

- Reflection: Angle of incidence (θ1/θi) = Angle of reflection (θ1’/θr).
- Refraction: Wave travelling from one medium into another.
- Snell’s law: v1/v2 = sinθ1/sinθ2 = n2/n1 = 1n2.
- Absolute refractive index = n = c/v = speed of light in vacuum/speed if light in medium.
- High n – optically dense medium.

- Absolute refractive index = n = c/v = speed of light in vacuum/speed if light in medium.

- Snell’s law: v1/v2 = sinθ1/sinθ2 = n2/n1 = 1n2.

It’s important to remember that, whenever refraction takes place, so does reflection!

- A ray will bend towards the normal if entering an optically denser medium.
- Plane: For reflection and refraction, the rays are always in the same plane.
- Reversibility of light: sinθ1/sinθ2 = 1n2 = 1/2n1 = sinθ2/sinθ1 = 2n1.

- The critical angle (θc): As the angle of incidence increases, the angle of refraction will approach 90º. At the angle of refraction 90º, the angle of incidence is called critical angle.
- sinθc = n2/n1.
- If angle of incidence (θ1 or θi) > critical angle (θc), there is total internal reflection.

### Diffraction

- Definition: “When a wave passes through a narrow slit, causing spread to bend and creating an interference pattern.”

- Occurrence: It takes place when the aperture (slit) ≤ wavelength. It is most evident when the aperture is significantly smaller than the wavelength.
- Quantities that…
- Remain constant: frequency, velocity and wavelength.
- Change: Amplitude reduces, since the energy is distributed over a larger area.

- Pattern of waves:

- Uses: CD/DVD or Electron microscope.

### Double-source interference

- Definition: “When two similar sources (with the same frequency) and coherent (with a constant phase relationship), emit waves that interfere with each other”.
- Path difference: The difference in distance of one specific point from the two sources.
- Path difference = ∆r =│S1P-S2P│, where S1P is the distance of source 1 to the specific point P and S2P is the distance of source 2 to the specific point P.
- Constructive interference: When ∆r = nλ, for n = 0, 1, 2, 3,…
- Destructive interference: When ∆r = (n + 1/2)λ, for n = 0, 1, 2, 3,…

Double-slit interference: Specific double-source interference, in which successive bright fringes are formed, as shown in the diagram below.

- Fringe spacing: s = λD/d, where D is the distance between the slits and the screen and d is the distance between the slits.
- Thomas Young experiment:

- Intensity graph: for negligible slit width.

### 4.5 Standing Waves

Definition: “When two travelling waves of equal amplitude and equal frequency travelling with the same speed in opposite directions are superposed, a standing/stationary wave is formed”.

Concepts

- Amplitude: Each particle has its own amplitude (A).
- Nodes: Points of destructive interference, i.e. zero amplitude.
- Anti-nodes: Points of constructive interference, i.e. maximum amplitude.
- Phase: Points between consecutive nodes are in phase.
- No energy transfer: A standing wave does not move horizontally, and thus, no energy is transferred and the shape does not change.

**Harmonics**

- Resonance: Systems, such as pipes and strings, only resonate at very specific frequencies, which are known as the harmonics.
- Harmonics have frequencies that are integral multiples of the first frequency, i.e the fundamental frequency. fn = n f1. They numbered according to n.

**Boundary conditions**

- Fixed boundary: Always a node, whose reflection causes a 180º phase change.
- Example: Walls or edge of a drum-head.

- Free boundary: Always an anti-node, whose reflection does not cause any phase change.
- Example: Tuning fork or air.

Cases

- Strings (Length = L): The waves reflect at the fixed ends, generating two identical waves travelling in opposite directions.
- End condition: node-node: λn = 2L/n or fn = nv/2L. Walls or edge of a drum-head.

- Pipes (Length = L): Medium is air – longitudinal waves!
- Longitudinal waves: Nodes are the centers of compression and rarefaction.
- End condition: totally closed: node-node. Equal treatment as strings (above).
- End condition: totally open: anti-node – anti-node. λn = 2L/n or fn = nv/2LEnd condition: partially open (one closed end): node – anti-node. λn = 4L/n or fn = nv/4L, only for odd harmonics, so n = 1, 3, 5…

Comparison between travelling and standing waves

Tip: Instead of trying to memorize the formulas for each specific condition or case in standing waves, try drawing the situation and then reaching the formula!

### WAVE MOTION

Wave motion is a type of motion in which the disturbance travels from one point of the medium to another but the particles of the medium do not travel from one point to another.For the propagation of wave, medium must have inertia and elasticity. These two properties of medium decide the speed of wave.There are two types of waves

- Mechanical waves : These waves require material medium for their propagation. For example : sound waves, waves in stretched string etc.
- Non-mechanical waves or electromagnetic waves : These waves do not require any material medium for their propagation. For example : light waves, x-rays etc.

There are two types of mechanical waves

- Transverse waves : In the transverse wave, the particles of medium oscillate in a direction perpendicular to the direction of wave propagation. Waves in stretched string, waves on the water surface are transverse in nature.
- Transverse wave can travel only in solids and surface of liquids.Transverse waves propagate in the form of crests and troughs. All electromagnetic waves are transverse in nature.

- Longitudinal waves : In longitudinal waves particles of medium oscillate about their mean position along the direction of wave propagation.
- Sound waves in air are longitudinal. These waves can travel in solids, liquids and gases.Longitudinal waves propagate through medium with the help of compressions and rarefactions.

### EQUATION OF A HARMONIC WAVE

Harmonic waves are generated by sources that execute simple harmonic motion.A harmonic wave travelling along the positive direction of x-axis is represented by where, y = displacement of the particle of the medium at a location x at time tA = amplitude of the waveλ = wavelengthT = time periodv = wave velocity in the mediumω = angular frequencyK = angular wave number or propagation constant.If the wave is travelling along the negative direction of x-axis then

DIFFERENTIAL EQUATION OF WAVE MOTION

RELATION BETWEEN WAVE VELOCITY AND PARTICLE VELOCITYThe equation of a plane progressive wave is … (i)The particle velocity … (ii)Slope of displacement curve or strain … (iii)Dividing eqn. (ii) by (iii), we geti.e., Particle velocity = – wave velocity × strain.Particle velocity changes with the time but the wave velocity is constant in a medium.

RELATION BETWEEN PHASE DIFFERENCE, PATH DIFFERENCE AND TIME DIFFERENCE

- Phase difference of 2π radian is equivalent to a path difference λ and a time difference of period T.
- Phase difference = × path difference
- Phase difference = × time difference
- Time difference = × path difference

### SPEED OF TRANSVERSE WAVES

- The speed of transverse waves in solid is given by

where η is the modulus of rigidity of the solid and ρ is the density of material.

- The speed of transverse waves on stretched string is given by

where T is the tension in the string and μ is the mass per unit length of the string.

### SPEED OF LONGITUDINAL WAVES

The speed of longitudinal waves in a medium of elasticity E and density ρ is given by For solids, E is replaced by Young’s modulus (Y)

For liquids and gases, E is replaced by bulk modulus of elasticity (B)

The density of a solid is much larger than that of a gas but the elasticity is larger by a greater factor.vsolid > vliquid > vgas

### SPEED OF SOUND IN A GAS

NEWTON’S FORMULAwhere P is the atmospheric pressure and ρ is the density of air at STP.

LAPLACE’S CORRECTIONwhere γ is the ratio of two specific heats Cp and Cv

### POWER AND INTENSITY OF WAVE MOTION

If a wave is travelling in a stretched string, energy is transmitted along the string.Power of the wave is given by where μ is mass per unit length.Intensity is flow of energy per unit area of cross section of the string per unit time.

### PRINCIPLE OF SUPERPOSITION OF WAVES

If two or more waves arrive at a point simultaneously then the net displacement at that point is the algebraic sum of the displacement due to individual waves.y = y1 + y2 + …………… + yn.where y1, y2 ………. yn are the displacement due to individual waves and y is the resultant displacement.

### INTERFERENCE OF WAVES

When two waves of equal frequency and nearly equal amplitude travelling in same direction having same state of polarisation in medium superimpose, then intensity is different at different points. At some points intensity is large, whereas at other points it is nearly zero.

Consider two waves y1 = A1sin (ωt – kx) and y2 = A2 sin (ωt – kx + φ) By principle of superposition y = y1 + y2 = A sin (ωt – kx + δ) where, A2 = A12 + A22 + 2A1A2 cos φ, and As intensity I ∝ A2So, resultant intensity I = I1 + I2 +

For constructive interference (maximum intensity) :Phase difference, φ = 2nπ and path difference = nλ where n = 0, 1, 2, 3, …⇒ Amax = A1 + A2 and Imax = I1 + I2 +

For destructive interference (minimum intensity) : Phase difference, φ = (2n + 1)π, and path difference = ; where n = 0, 1, 2, 3, …⇒ Amin = A1 – A2 and Imin = I1 + I2 –

RESULTS

- The ratio of maximum and minimum intensities in any interference wave form.

- Average intensity of interference in wave form :

Put the value of Imax and Iminor Iav = I1 + I2If A = A1 = A2 and I1 = I2 = Ithen Imax = 4I, Imin = 0 and Iav = 2I

- Condition of maximum contrast in interference wave form

A1 = A2 and I1 = I2then Imax = 4I and Imin = 0For perfect destructive interference we have a maximum contrast in interference wave form.

REFLECTION OF WAVESA mechanical wave is reflected and refracted at a boundary separating two media according to the usual laws of reflection and refraction.When sound wave is reflected from a rigid boundary or denser medium, the wave suffers a phase reversal of π but the nature does not change i.e., on reflection the compression is reflected back as compression and rarefaction as rarefaction.When sound wave is reflected from an open boundary or rarer medium, there is no phase change but the nature of wave is changed i.e., on reflection, the compression is reflected back as rarefaction and rarefaction as compression.

KEEP IN MEMORY

For a wave, v = f λ

The wave velocity of sound in air

Particle velocity is given by. It changes with time. The wave velocity is the velocity with which disturbances travel in the medium and is given by .

When a wave reflects from denser medium the phase change is π and when the wave reflects from rarer medium, the phase change is zero.

In a tuning fork, the waves produced in the prongs is transverse whereas in the stem is longitudinal.

A medium in which the speed of wave is independent of the frequency of the waves is called non-dispersive. For example air is a non-dispersive medium for the sound waves.

Transverse waves can propagate in medium with shear modulus of elasticity e.g., solid whereas longitudinal waves need bulk modulus of elasticity hence can propagate in all media solid, liquid and gas.

#### ENERGY TRANSPORTED BY A HARMONIC WAVE ALONG A STRING

Kinetic energy of a small element of length dx is where μ = mass per unit lengthand potential energy stored

### BEATS

When two wave trains slightly differing in frequencies travel along the same straight line in the same direction, then the resultant amplitude is alternately maximum and minimum at a point in the medium. This phenomenon of waxing and waning of sound is called beats.

Let two sound waves of frequencies n1 and n2 are propagating simultaneously and in same direction. Then at x=0y1 = A sin 2π n1t, and y2 = A sin 2π n2t,For simplicity we take amplitude of both waves to be same.By principle of superposition, the resultant displacement at any instant is y = y1 + 2 = 2A cos 2π nAt sin 2π navtwhere , ⇒ y = Abeat sin 2π navt ………………(i)It is clear from the above expression (i) that

- Abeat = 2A cos 2πnAt, amplitude of resultant wave varies periodically as frequency

A is maximum when A is minimum when

- Since intensity is proportional to amplitude i.e.,

For Imax cos 2π nAt = ± 1 For Imini.e., 2π nAt = 0,π, 2π 2π nAt = π/2, 3π/2i.e., t = 0, 1/2nA, 2/2nA t = 1/4nA, 3/4nA……. So time interval between two consecutive beat is

Number of beats per sec is given by

So beat frequency is equal to the difference of frequency of two interfering waves.To hear beats, the number of beats per second should not be more than 10. (due to hearing capabilities of human beings)

#### FILING/LOADING A TUNING FORK

On filing the prongs of tuning fork, raises its frequency and on loading it decreases the frequency.

- When a tuning fork of frequency ν produces Δν beats per second with a standard tuning fork of frequency ν0, then

If the beat frequency decreases or reduces to zero or remains the same on filling the unknown fork, then

- If the beat frequency decreases or reduces to zero or remains the same on loading the unknown fork with a little wax, then

If the beat frequency increases on loading, then

### DOPPLER EFFECT

When a source of sound and an observer or both are in motion relative to each other there is an apparent change in frequency of sound as heard by the observer. This phenomenon is called the Doppler’s effect.Apparent change in frequency

- When source is in motion and observer at rest
- when source moving towards observer

- when source moving away from observer

Here V = velocity of soundVS = velocity of sourceν0 = source frequency.

- When source is at rest and observer in motion
- when observer moving towards source

- when observer moving away from source and

V0 = velocity of observer.

- when observer moving away from source and

- When source and observer both are in motion
- If source and observer both move away from each other.

- If source and observer both move towards each other.

When the wind blows in the direction of sound, then in all above formulae V is replaced by (V + W) where W is the velocity of wind. If the wind blows in the opposite direction to sound then V is replaced by (V – W).

KEEP IN MEMORY

- The motion of the listener causes change in number of waves received by the listener and this produces an apparent change in frequency.
- The motion of the source of sound causes change in wavelength of the sound waves, which produces apparent change in frequency.
- If a star goes away from the earth with velocity v, then the frequency of the light emitted from it changes from ν to ν’.

ν’ = ν (1–v/c), where c is the velocity of light and where is called Doppler’s shift.

If wavelength of the observed waves decreases then the object from which the waves are coming is moving towards the listener and vice versa.

### STATIONARY OR STANDING WAVES

When two progressive waves having the same amplitude, velocity and time period but travelling in opposite directions superimpose, then stationary wave is produced.Let two waves of same amplitude and frequency travel in opposite direction at same speed, then y1 = A sin (ωt –kx) and y2 = A sin (ωt + kx)By principle of superpositiony = y1 + y2 = (2A cos kx) sin ωt …(i)y = AS sinωtIt is clear that amplitude of stationary wave As vary with position

- As = 0, when cos kx = 0 i.e., kx = π/2, 3π/2…………

i.e., x = λ/4, 3λ/4……………….[as k = 2π/λ]

These points are called nodes and spacing between two nodes is λ/2.

- As is maximum, when cos kx is max

i.e., kx = 0, π , 2π, 3π i.e., x = 0, λl/2, 2λ/2…. It is clear that antinode (where As is maximum) are also equally spaced with spacing λ/2.

- The distance between node and antinode is λ/4 (see figure)

KEEP IN MEMORY

When a string vibrates in one segment, the sound produced is called fundamental note. The string is said to vibrate in fundamental mode.

The fundamental note is called first harmonic, and is given by, where v = speed of wave.

If the fundamental frequency be then , , … are respectively called second third, fourth … harmonics respectively.

If an instrument produces notes of frequencies …. where ….., then is called first overtone, is called second overtone, is called third overtone … so on.

Harmonics are the integral multiples of the fundamental frequency. If ν0 be the fundamental frequency, then nν0 is the frequency of nth harmonic.

Overtones are the notes of frequency higher than the fundamental frequency actually produced by the instrument.

In the strings all harmonics are produced.

#### STATIONARY WAVES IN AN ORGAN PIPE

In the open organ pipe all the harmonics are produced.In an open organ pipe, the fundamental frequency or first harmonic is , where v is velocity of sound and l is the length of air column

[see fig. (a)] (a)

, (b) , (c) ,

Similarly the frequency of second harmonic or first overtone is [see fig (b)], Similarly the frequency of third harmonic and second overtone is [(see fig. (c)] Similarly ….In the closed organ pipe only the odd harmonics are produced. In a closed organ pipe, the fundamental frequency (or first harmonic) is (see fig. a)(a)

(b) (c)

Similarly the frequency of third harmonic or first overtone (IInd harmonic absent) is (see fig. b)Similarly ……..

End CorrectionIt is observed that the antinode actually occurs a little above the open end. A correction is applied for this which is known as end correction and is denoted by e.

- For closed organ pipe : l is replaced by l + e where e = 0.3D, D is the diameter of the tube.
- For open organ pipe : l is replaced by l + 2e where e = 0.3D

In resonance tube, the velocity of sound in air given by v = 2v (l2–l1)where ν = frequency of tuning fork, ll = 1st resonating length, l2 = 2nd resonating length.

#### RESONANCE TUBE

It is used to determine velocity of sound in air with the help of a tuning fork of known frequency.

Let l1 and l2 are lengths of first and second resonances then and Speed of sound in air is where υ is the frequency

For vibrating strings/open organ pipe

For closed organ pipe

#### COMPARISON OF PROGRESSIVE (OR TRAVELLING) AND STATIONARY (OR STANDING) WAVE

#### COMPARATIVE STUDY OF INTERFERENCE, BEATS AND STATIONARY WAVE

### CHARACTERISTICS OF SOUND

Musical sound – consists of quick, regular and periodic succession of compressions and rarefactions without a sudden change in amplitude.

Noise – consists of slow, irregular and a periodic succession of compressions and rarefactions that may have sudden changes in amplitude.

Pitch, loudness and quality are the characteristics of musical sound.

- Pitch depends on frequency
- loudness depends on intensity
- quality depends on the number and intensity of overtones

Interval – The ratio of the frequencies of the two notes is called the interval between them. For example interval between two notes of frequencies 512 Hz and 1024 Hz is 1 : 2 (or 1/2).Two notes are said to be in unison if their frequencies are equal, i.e., if the interval between them is 1 : 1. Some other common intervals, found useful in producing musical sound are the following:Octave (1 : 2), majortone (8 : 9), minortone (9 : 10) and semitone (15 : 16)

Major diatonic scale – It consists of eight notes. The consecutive notes have either of the following three intervals. They are 8 : 9 ; 9 : 10 and 15 : 16.

#### ACOUSTICS

The branch of physics that deals with the process of generation, reception and propagation of sound is called acoustics.Acoustics may be studied under the following three subtitles.

- Electro acoustics. This branch deals with electrical sound production with music.
- Musical acoustics. This branch deals with the relationship of sound with music.
- Architectural acoustics. This branch deals with the design and construction of buildings.

#### REVERBERATION

Multiple reflections which are responsible for a series of waves falling on listener’s ears, giving the impression of a persistence or prolongation of the sound are called reverberations.The time gap between the initial direct note and the reflected note upto the minimum audibility level is called reverberation time.

Sabine Reverberation Formula for TimeSabine established that the standard period of reverberation viz., the time that the sound takes to fall in intensity by 60 decibels or to one millionth of its original intensity after it was stopped, is given by where V = volume of room, = α1 S1 + α2 S2 + ….S1, S2 …. are different kinds of surfaces of room and

α1 , α2 …. are their respective absorption coefficient.The above formula was derived by Prof C. Sabine.

#### SHOCK WAVES

The waves produced by a body moving with a speed greater than the speed of sound are called shock waves. These waves carry huge amount of energy. It is due to the shock wave that we have a sudden violent sound called sonic boom when a supersonic plane passes by.The rate of speed of the source to that of the speed of sound is called mach number.

#### INTENSITY OF SOUND

The sound intensities that we can hear range from 10–12 Wm–2 to 103 Wm–2. The intensity level β, measured in terms of decibel (dB) is defined as where I = measured intensity, I0 = 10–12 Wm–1 At the threshold β = 0At the max

#### LISSAJOUS FIGURES

When two simple harmonic waves having vibrations in mutually perpendicular directions superimpose on each other, then the resultant motion of the particle is along a closed path, called the Lissajous figures. These figures can be of many shapes depending on

- ratio of frequencies or time periods of two waves
- ratio of amplitude of two waves
- phase difference between two waves.