Home / IB DP Physics- Topic 4. Waves: 4.4 Wave behavior Study Notes

IB DP Physics- Topic 4. Waves: 4.4 Wave behavior Study Notes

4.4 Wave behaviour

Essential Idea:
Waves interact with media and each other in a number of ways that can be unexpected and useful

Understandings:

• Reflection and refraction
• Snell’s law, critical angle and total internal reflection
• Diffraction through a single-slit and around objects
• Interference patterns
• Double-slit interference
• Path difference

Applications and Skills:

• Sketching and interpreting incident, reflected and transmitted waves at boundaries between media
• Solving problems involving reflection at a plane interface
• Solving problems involving Snell’s law, critical angle and total internal reflection
• Determining refractive index experimentally
• Qualitatively describing the diffraction pattern formed when plane waves are incident normally on a single-slit
• Quantitatively describing double-slit interference intensity patterns

Data booklet reference:

• $$\frac{n_1}{n_2}=\frac{sin\theta _2}{sin\theta _1}=\frac{v_2}{v_1}$$
• $$s=\frac{\lambda D}{d}$$
• Constructive interference: path difference = nλ
• Destructive interference: path difference = $$(n+\frac{1}{2})\lambda$$

REFRACTION

Whenever a wave is bounced back into same medium at an interface reflection is said to have occurred. Transmission of a wave into the second medium at an interface is called refraction. When a ray of light is passing from denser to rarer medium, it bends away from the normal and when passing from rarer to denser medium, it bends towards the normal.
• When a ray of light passing from one medium to another medium frequency and phase do not change while wavelength and velocity changes.
• Twinkling of stars, appearance of sun before actual sunrise and after actual sunset etc. are due to atmospheric refraction.

LAWS OF REFRACTION

• Snell’s Law : When a light ray is incident on a surface separating two transparent media, the ray bends at the time of changing the medium.
i.e. ,
where i = angle of incidence
r = angle of refraction
v1 = vel. of light in 1st medium
v2 = vel. of light in 2nd medium
or 1μ2 = refractive index of 2nd medium w.r.t. the 1st medium.
μ1 = refractive index of 1st medium w.r.t vacuum (or air)
μ2 = refractive index of 2nd medium w.r.t vacuum (or air)
• The incident ray, the normal and the refracted ray at the interface all lie in the same plane.

REFRACTIVE INDEX OF THE MEDIUM

REFRACTIVE INDEX OF SECOND MEDIUM W.R.T. FIRST MEDIUM

ABSOLUTE REFRACTIVE INDEX OF MEDIUM (n OR µ)

Refractive index is the relative property of two media. If the first medium carrying the incident ray is a vacuum, then the ratio is called the ‘absolute refractive index of the second medium’. The relative refractive index of any two media is equal to the ratio of their absolute refractive indices.
Therefore, if the absolute refractive index of medium 1 and 2 be n1 and n2 respectively, then the refractive index of medium 2 with respect to medium 1 is
n1 sin i = n2 sin r
According to cauchy’s formula
where, A and B are  cauchy’s constant.
λred > λviolet so,   μred < μviolet
• For three mediums 1, 2 and 3 due to successive refraction.
1n2 × 2n3 × 3n1 = 1
• For two mediums, n1 and n2 are refractive indices with respect to vacuum, the incident and  emergent rays are parallel then
n1 sinφ1 = n2 sinφ2.

FACTORS AFFECTING REFRACTIVE INDEX
• Nature of the medium
• Wavelength
• Temperature of the medium-with increase in temperature, refractive index of medium decreases.

TRANSMISSION OF WAVE

• The equation of the wave refracted or transmitted to the next medium is given by : y = A´´ sin. This is independent of the nature (rarer/denser) of the medium. The wave is not inverted.
• The amplitude (A´´) of the transmitted wave is less than that (A) of the incident wave.
• The angular frequency remains unchanged. However the wave number changes. Note that the phase of the transmitted wave is and that of the incident wave is (ωt – kx).
• The compression or rarefaction are transmitted as such and same is the case with the crest or trough.
The wave velocity (vp), the angular frequency (ω) and the wave number (k) are related as vp = ω/k = nλ. Let the wave velocity in the medium to which the wave is transmitted be v’p = ω/k´ = nλ’.
Frequency is the characteristic of the source while wavelength is the characteristic of the medium. when monochromatic light travels from one medium to another its speed changes so its wavelength changes but frequency v remains unchanged.
• If second medium is denser, in comparison to first medium (i.e. μ2 > μ1), then from Snell’s law
here μ21 so
⇒ k1 < k2 and λ1 > λ2.
It means that if ray goes from rarer medium to denser medium (i.e. from first medium to second medium), then wavenumber increases & wavelength decreases.
• If second medium is rarer in comparison to first medium, then from Snell’s law
here μ2 < μ1 so
⇒ k1 > k2 and λ1 < λ2.
It means that when ray goes from denser to rarer medium, then wave number decreases & wavelength increases.
• No change in wave number k occurs on reflection.

WAVEFRONT

The locus of all particles of the medium vibrating in the same phase at a given instant is called a wavefront.
Depending on the shape of source of light, wavefront can be of three types.
• Spherical wavefront : A spherical wavefront is produced by a point source of light. This is because the locus of all such points which are equidistant from the point source will be a sphere. Spherical wavefronts are further divided into two headings: (i) converging spherical and (ii) diverging spherical wavefront.
• Cylindrical wavefront : When the source of light is linear in shape such as a slit, the cylindrical wavefront is produced. This is because all the points equidistant from a line source lie on the surface of a cylinder.
• Plane wavefront : A small part of a spherical or cylindrical wavefront due to a distant source will appear plane and hence it is called plane wave-front. The wavefront of parallel rays is a plane wavefront.

HUYGENS WAVE THEORY

(Geometrical method to find the secondary wavefront)
• Each point source of light is a centre of disturbance from which waves spread in all directions.
• Each point on primary wavelets acts as a new source of disturbance and produces secondary wavelets which travel in space with the speed of light.
• The forward envelope of the secondary wavelets at any instant gives the new wavefront.

• In a homogeneous medium the wavefront is always perpendicular to the direction of wave propagation.

Note:- With the help of Huygen’s wave theory, law of reflection and refraction, total internal reflection and dispersion can be explained easily. This theory also explain interference, diffraction and polarization successfully.

DRAWBACKS OF HUYGENS WAVE THEORY

• This theory cannot explain photoelectric effect, Compton, and Raman effect.
• Hypothetical medium in vacuum is not true imagination.
• The theory predicted the presence of back wave, which proved to be failure.

REFLECTION AND REFRACTION OF PLANE WAVES USING HUYGENS PRINCIPLE

REFLECTION ON THE BASIS OF WAVE THEORY

According to Huygens principle, every point on AB is a source of secondary wavelets. Let the secondary wavelets from B strike reflecting surface M1M2 at A′ in t seconds.
… (i)
where c is the velocity of light in the medium.
The secondary wavelets from A will travel the same distance c × t in the same time. Therefore, with A as centre and c × t as radius, draw an arc B′, so that
AB′ = c × t … (ii)
A′B′ is the true reflected wavefront.
angle of incidence, i =
and angle of reflection, r =
In Δs AA′B and AA′B′,
AA′ is common,  and
∴ Δs are congruent ∴ … (iii)
which is the first law of reflection.

Further, the incident wavefront AB, the reflecting surface M1M2 and the reflected wavefront A′B′ are all perpendicular to the plane of the paper. Therefore, incident ray, normal to the mirror M1M2 and reflected ray all lie in the plane of the paper. This is second law of reflection.

REFRACTION ON THE BASIS OF WAVE THEORY

XY is a plane surface that separates a denser medium of refractive index µ from a rarer medium. If c1 is velocity of light in rarer medium and c2 is velocity of light in denser medium, then by definition.
µ =   … (iv)
AB is a plane wave front incident on XY at = . 1, 2, 3 are the corresponding incident rays normal to AB.
According to Huygens principle, every point on AB is a source of secondary wavelets. Let the secondary wavelets from B strike XY at A′ in t seconds.
BA′ = c1 × t … (v)

The secondary wavelets from A travel in the denser medium with a velocity c2 and would cover a distance (c2 × t) in t seconds. A′B′ is the true refracted wavefront. Let r be the angle of refraction. As angle of refraction is equal to the angle which the refracted plane wavefront A′B′ makes with the refracting surface AA′, therefore, .
Let, angle of refraction.
In  ΔAA′B,
In Δ AA′B′,
[using (iv)]
Hence       … (vi)
which proves Snell’s law of refraction.

It is clear from fig. that the incident rays, normal to the interface XY and refracted rays, all lie in the same plane (i.e., in the plane of the paper). This is the second law of refraction.
Hence laws of refraction are established on the basis of wave theory.

KEEP IN MEMORY
1. In 1873, Maxwell showed that light is an electromagnetic wave i.e. it propagates as transverse non-mechanical wave at speed c in free space given by
1. There are some phenomenon of light like photoelectric effect, Compton effect, Raman effect etc. which can be explained only on the basis of particle nature of light.
2. Light shows the dual nature i.e. particle as well as wave nature of light. But the wave nature and particle nature both cannot be possible simultaneously.
3. Interference and diffraction are the two phenomena that can be explained only on the basis of wave nature of light.

INTERFERENCE OF LIGHT WAVES AND YOUNG’S DOUBLE SLIT EXPERIMENT

The phenomenon of redistribution of light energy in a medium on account of superposition of light waves from two coherent sources is called interference of light waves.

Young performed the experiment by taking two coherent sources of light. Two source of light waves are said to be coherent if the initial phase difference between the waves emitted by the source remains constant with time.

The rays of light from two coherent sources S1 and S2 superpose each other on the screen forming alternately maxima and minima (constructive and destructive interference).
Let the equation of waves travelling from are
…(1)
…(2)
where A1 & A2 are amplitudes of waves starting  from S1 & S2 respectively. These two waves arrive at P by traversing different distances S2P & S1P. Hence they are superimposed with a phase difference (at point P) given by
….(3)
where (from fig)  =
Similarly,
so,   ….(4)

(A) CONDITIONS FOR MAXIMUM & MINIMUM INTENSITY

CONDITIONS FOR MAXIMUM INTENSITY OR CONSTRUCTIVE INTERFERENCE
If phase difference
δ = 0, 2π,  4π – – – 2nπ
or, path difference
then resultant intensity at point P due two waves emanating from S1 & S2 is

or
or  ….(5)
It means that resultant intensity is greater than the sum of individual intensity ( where A is the amplitude of resultant wave at point P).

CONDITIONS FOR MINIMUM INTENSITY OR DESTRUCTIVE INTERFERENCE
If phase difference,
or,  path difference
then resultant intensity at point P is
or
or   …(6)
It means that resultant intensity I is less than the sum of individual intensities. Now as the position of point P on the screen changes, then the path difference at point P due to these two waves also changes & intensity alternately becomes maximum or minimum. These bright fringes (max. intensity) & dark fringes (min. intensity) make an interference pattern.
It must be clear that there is no loss of energy (at dark fringe) & no gain of energy (at bright fringe), but only there is a redistribution of energy.
The shape of fringe obtained on the screen is approximately linear.

(B) POSITION OF FRINGE

If Δ = S2P – S1P = nλ, then we obtain bright fringes at point P on the screen and it corresponds to constructive interference. So from equation (4) the position of nth bright fringe
or       …(7)
(Position of nth bright fringe)

If, then we obtain dark fringe at point P on the screen and corresponds to destructive interference. So from equation (4), the position of, nth dark fringe is
or  …(8)
(Position of nth dark fringe)

(C) SPACING OR FRINGE WIDTH

Let yn and yn+1 are the distance of nth and (n+1)th bright fringe from point O then
&
So spacing β between nth and (n+1)th bright fringe is
…(9)
Since it is independent of n, so fringe width or spacing between any two consecutive bright fringes is same.
Similarly the fringe width between any two consecutive dark fringes is
…(10)

(D) CONDITIONS FOR SUSTAINED INTERFERENCE

• The two sources should be coherent i.e. they should have a constant phase difference between them.
• The two sources should give light of same frequency (or wavelength).
• If the interfering waves are polarized, then they must be in same state of polarization.

(E) CONDITIONS FOR GOOD OBSERVATION OF FRINGE

• The distance between two sources i.e. d should be small.
• The distance of screen D from the sources should be quite large.
• The two interfering wavefronts must intersect at a very small angle.

(F) CONDITIONS FOR GOOD CONTRAST OF FRINGE

• Sources must be monochromatic i.e. they emit waves of single wavelength.
• The amplitude of two interfering waves should be equal or nearly equal.
• Both sources must be narrow.
• As Intensity I is directly proportional to the square of amplitude, hence Intensity of resultant wave at P,
, then
If I1 = I2 = I0, then Imax = 4I0
, if I1 = I2 = I0, then Imin = 0
• Angular fringe-width
• The width of all interference fringes are same. Since fringe width β  is proportional to λ, hence fringes with red light are wider than those for blue light.
• If the interference experiment is performed in a medium of refractive index μ instead of air, the wavelength of light will change from λ to .
i.e.
• If a transparent sheet of refractive index μ and thickness t is introduced in one of the paths of interfering waves, then due to its presence optical path will become μt instead of t. Due to this a given fringe from present position shifts to a new position. So the lateral shift of the fringe,
• In Young’s double slit experiment (coherent sources in phase): Central fringe is a bright fringe. It is on the perpendicular bisector of coherent sources. Central fringe position is at a place where two waves having equal phase superpose.
• Young’s experiment with the white light will give white central fringe flanked on either side by coloured bands.

COHERENCE

The phase relationship between two light waves can vary from time to time and from point to point in space. The property of definite phase relationship is called coherence.

TEMPORAL COHERENCE

A light wave (photon) is produced when an excited atom goes to the ground state and emits light.
• The duration of this transition is about 10–9 to 10–10 sec. Thus the emitted wave remains sinusoidal for this much time. This time is known as coherence time ().
• Definite phase relationship is maintained for a length called coherence length.
For neon ,   and L = 0.03 m.
For cadmium , and L = 0.3 m
For Laser  sec and L = 3 km.
• The spectral lines width is related to coherence length L and coherence time τc.
or

SPATIAL COHERENCE

Two points in space are said to be spatially coherence if the waves reaching there maintains a constant phase difference. Points P and Q are at the same distance from S, they will always be having the same phase.
Points P and P′ will be spatially coherent if the distance between P and P′ is much less than the coherence length i.e.

METHODS OF OBTAINING COHERENT SOURCES

Two coherent sources are produced from a single source of light by two methods :
• By division of wavefront
• By division of amplitude.

DIVISION OF WAVEFRONT
The wavefront emitted by a narrow source is divided in two parts by reflection, refraction or diffraction. The coherent sources so obtained are imaginary. Example : Fresnel’s biprism, Llyod’s mirror, Young’s double slit, etc.

DIVISION OF AMPLITUDE
In this arrangement light wave is partly reflected (50%) and partly transmitted (50%) to produced two light rays. The amplitude of wave emitted by an extended source of light is divided in two parts by partial reflection and partial refraction. The coherent sources obtained are real and are obtained in Newton’s rings, Michelson’s interferometer, etc.

INCOHERENCE OF TWO CONVENTIONAL LIGHT SOURCES

Let two conventional light sources L1 and L2 (like two sodium lamps or two monochromatic bulbs) illuminate two pin holes S1 and S2. Then we will find that no interference pattern is seen on the screen.
The reason is as follows : In conventional light source, light comes from a large number of independent atoms, each atom emitting light for about 10–9 seconds i.e., light emitted by an atom is essentially a pulse lasting for only 10–9 seconds.
Even if all the atoms were emitting light pulses under similar conditions, waves from different atoms would differ in their initial phases. Consequently light coming out from the holes S1 and S2 will have a fixed phase relationship only for 10–9 sec. Hence any interference pattern formed on the screen would last only for 10–9 sec. (a billionth of a second), and then the pattern will change. The human eye can notice intensity changes which last at least for a tenth of a second and hence we will not be able to see any interference pattern. Instead due to rapid changes in the pattern, we will only observe a uniform intensity over the screen.

LLOYD’S MIRROR

The two sources are slit S (parallel to mirror) and its virtual image S’.
• If screen is moved so that, point O touches the edge of glass plate, the geometrical path difference for two wave trains is zero. The phase change of π radian on reflection at denser medium causes a dark fringe to be formed.
• The fringe width remains unchanged on introduction of transparent film.
• If the film is placed in front of upper slit S1, the fringe pattern will shift upwards. On the other hand if the film is placed in front of lower slit S2, the fringe pattern shifts downwards.
• This interference pattern is frequently seen in a ripple tank when one uses a wave train to demonstrate the law of reflection.
• In this case, fringe width
Optical path : (Equivalent path in vacuum or air) In case of medium of refractive index μ and thickness t, the optical path = μt.

INTERFERENCE IN THIN FILMS

We are familiar with the colours produced by a thin film of oil on the surface of water and also by the thin film of a soap bubble. Hooke observed such colours in thin films of mica and similar thin transparent plates. Young was able to explain the phenomenon on the basis of interference between light reflected from the top and bottom surface of a thin film. It has been observed that interference in the case of thin films takes place due to
• reflected light
• transmitted light

INTERFERENCE DUE TO REFLECTED LIGHT

From the figure, the optical path difference between the reflected ray (AT) from the top surface and the reflected ray (CQ) from the bottom surface can be calculated. Let it be x, then
On simplification, we get x = 2μt cos r
• If , where n = 0,1,2, …………..then constructive interference takes place and the film appears bright.
• If , where n = 0, 1, 2, 3,………… then destructive interference takes place and the film appears dark.

INTERFERENCE DUE TO TRANSMITTED LIGHT

The optical path difference between the reflected ray (DQ) and the transmitted ray (NR) is given by
On simplification, we get x = 2μt cos r
1. If , where n = 0, 1, 2, 3, ………….then constructive interference takes place and the film appears bright.
2. If , n = 0, 1, 2, ……. then destructive interference takes place and the film appears dark.

NEWTON’S RINGS

Newton observed the formation of interference rings when a plano-convex lens is placed on a plane glass plate. When viewed with white light, the fringes are coloured while with monochromatic light, the fringes are bright and dark. These fringes are produced due to interference between the light reflected from the lower surface of the lens and the upper surface of the glass plate. Interference can also take place due to transmitted light.

NEWTON’S RINGS BY REFLECTED LIGHT

Here, interference takes place due to reflected light. Therefore,
for bright rings,……
for dark rings, ……
Proceeding further, we get the radius of rings as follows:
for bright rings,
for dark rings,
where R = radius of curvature of lens.

Note:-
• The centre is dark and alternately dark and bright rings are produced.
• While counting the order of the dark rings 1, 2, 3, etc. the central ring is not counted. Therefore,
for 1st dark ring, n = 1 and
for 2nd dark ring, n = 2 and

NEWTON’S RINGS BY TRANSMITTED LIGHT

Here, interference takes place due to transmitted light.
Therefore,
for bright rings,
for dark rings,
Proceeding further, we get

Note:-
• The centre is bright and alternately bright and dark rings are obtained.
• The ring pattern due to reflected light is just opposite to that of transmitted light.

KEEP IN MEMORY
• If Dn and Dn + m be the diameters of n th and (n + m)th dark rings then the wavelength of light used is given by
where, R is the radius of curvature of the lens.
• If  Dn = diameter of nth dark ring when air is present between the glass plate and the lens
Dn+m =diameter of (n+m)th dark ring when air is present between the glass plate and the lens
D′n = diameter of n th dark ring when a liquid is poured between the plate and the lens
D′n+m = diameter of (n+m)th dark ring when a liquid is poured between the plate and the lens
Then the refractive index of the liquid is given by

DIFFRACTION

When a wave is obstructed by an obstacle, the rays bend round the corner. This phenomenon is known as diffraction.

FRAUNHOFER DIFFRACTION BY SINGLE SLIT

In Fraunhofer diffraction experiment, the source and the screen are effectively at infinite distance from the diffracting element.
In single slit diffraction, imagine aperture to be divided into two equal halves. Secondary sources in these two halves give first minima at b sin θ = λ
In general, b sin θ = nλ for minima and, for maxima.
• The points of the maximum intensity lie nearly midway between the successive minima. The amplitude E0‘ of the electric field at a general point P is
where and
E0 = amplitude at the point P0 i.e. at θ = 0
The intensity at a general point P is given as
• The graph for the variation of intensity as a function of sinθ is as follows :
• The width of the central maxima is and angular width of central maxima is.

FRAUNHOFER DIFFRACTION BY A CIRCULAR APERTURE

• The 1st dark ring is formed by the light diffracted from the circular aperture at an angle θ with the axis where
where λ = wavelength of light used, b = diameter of circular aperture
• If the screen is at a distance D (D >> b) from the circular aperture, the radius of the 1st dark ring is,
• If the light transmitted by the hole is converged by a converging lens at the screen placed at the focal plane of the lens, the radius of the 1st dark ring is
For plane transmission diffraction grating
(a + b) sin θn = nλ  for maxima, where a = width of transparent portion, b = width of opaque portion.

POLARISATION

An ordinary source such as bulb consists of a large number of waves emitted by atoms or molecules in all directions symmetrically. Such light is called unpolarized light (see fig – a)
If we confine the direction of wave vibration of electric vector  in one direction perpendicular to direction of wave propagation, then such type of light is called plane polarised or linearly polarised (with the help of polaroids or Nicol prism). The phenomenon by which, we restrict the vibrations of wave in a particular direction (see fig-b) ⊥ to direction of wave propagation is called polarization.
The plane of vibration is that which contains the vibrations of electric vector and plane of polarisation is perpendicular to the plane of vibration
Tourmaline and calcite polarises an e.m. wave passing through it.

POLARIZATION BY REFLECTION (BREWSTER’S LAW)

During reflection of a wave, we obtain a particular angle called angle of polarisation, for which the reflected light is completely plane polarised.
μ = tan (ip)
where, ip = angle of incidence, such that the reflected and refracted waves are perpendicular to each other.

LAW OF MALUS

If the electric vector is at angle θ with the transmission axis, light is partially transmitted. The intensity of transmitted light is
I = I0 cos2θ where I0 is the intensity when the incident electric vector is parallel to the transmission axis.

Polarization can also be achieved by scattering of light
• Plane polarized : oscillating electric field is in a single plane.
• Circularly polarized : tip of oscillating electric field describes a circle.
• Elliptically polarized : tip of oscillating electric field describes an ellipse.

RESOLVING POWER OF AN OPTICAL INSTRUMENT

The resolving power of an optical instrument, is its ability to distinguish between two closely spaced objects.

Diffraction occurs when light passes through the circular, or nearly circular, openings that admit light into cameras, telescopes, microscopes, and human eyes. The resulting diffraction pattern places a natural limit on the resolving power of these instruments.
For example, for normal vision, the limit of resolution of normal human eye is ~0.1 mm from 25 cm. (i.e., distances less than 0.1 mm cannot be resolved). For optical microscope the limit of resolution ~ 10–5 cm and for electron microscope ~5 Å or less.
The limit of resolution of a microscope where a is the aperture of the microscope.

DOPPLER’S EFFECT FOR LIGHT WAVES

When the source moves towards the stationary observer or the observer moves towards the source, the apparent frequency.

When the source moves away from the stationary observer or vice-versa, the apparent frequency

where ν´ = apparent frequency, ν = active frequency
v = velocity of source, c = velocity of light
But in both cases, the relative velocity v is small.

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 t
A = amplitude of the wave
λ = wavelength
T = time period
v =  wave velocity in the medium
ω = angular frequency
K = 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 VELOCITY
The 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 get
i.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

Longitudinal Waves

In a longitudinal wave the particle displacement is parallel to the direction of wave propagation. The animation at right shows a one-dimensional longitudinal plane wave propagating down a tube. The particles do not move down the tube with the wave; they simply oscillate back and forth about their individual equilibrium positions. Pick a single particle and watch its motion. The wave is seen as the motion of the compressed region (ie, it is a pressure wave), which moves from left to right.

The second animation at right shows the difference between the oscillatory motion of individual particles and the propagation of the wave through the medium. The animation also identifies the regions of compression and rarefaction.

The P waves (Primary waves) in an earthquake are examples of Longitudinal waves. The P waves travel with the fastest velocity and are the first to arrive

Transverse Waves

In a transverse wave the particle displacement is perpendicular to the direction of wave propagation. The animation below shows a one-dimensional transverse plane wave propagating from left to right. The particles do not move along with the wave; they simply oscillate up and down about their individual equilibrium positions as the wave passes by. Pick a single particle and watch its motion.

The S waves (Secondary waves) in an earthquake are examples of Transverse waves. S waves propagate with a velocity slower than P waves, arriving several seconds later.

Standing and travelling waves

The main differences between a standing wave and a travelling wave are summarised below.

• In a travelling wave, the disturbance produced in a region propagates with a definite velocity but in a standing wave, it is confined to the region where it
is produced.
• In a travelling wave, the motion of all the particles are similar in nature. In a standing wave, different particles move with different amplitudes.
• In a standing wave, the particles at nodes always remain in rest. In travelling waves, there is no particle which always remains in rest.
• In a standing wave, all the particles cross their mean positions together. In a travelling wave, there is no instant when all the particles are at the mean positions together.
• In a standing wave, all the particles between two successive nodes reach their extreme positions together, thus moving in phase. In a travelling wave, the phases of nearby particles are always different.
• In a travelling wave, energy is transmitted from one region of space to other but in a standing wave, the energy of one region is always confined in that region.

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

The regions of higher and lower pressures in sound waves are called compressions and rarefactions, respectively.
Sound waves can have any frequency; the human ear can hear sounds between about 20 Hz and 20,000 Hz. The frequencies of ultrasound waves
are above our range of hearing.
The distance between compression and the next rarefaction of a longitudinal wave is $$\frac{\lambda }{2}$$
The graph shows how the displacement of particles vary with distance from the source. Point B is the centre of a compression.

Parameters of water waves

SPEED OF SOUND IN A GAS

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

LAPLACE’S CORRECTION
where 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 ∝ A2
So, 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
1. The ratio of maximum and minimum intensities in any interference wave form.
1. Average intensity of interference in wave form :

Put the value of  Imax and Imin
or Iav = I1 + I2
If A = A1 = A2 and I1 = I2 = I
then Imax = 4I, Imin = 0  and Iav = 2I
1. Condition of maximum contrast in interference wave form
A1 = A2 and  I1 = I2
then Imax = 4I and Imin = 0
For perfect destructive interference we have a maximum contrast in interference wave form.

REFLECTION OF WAVES
A 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
1. For a wave, v = f λ
2. The wave velocity of sound in air
1. 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 .
2. 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.
3. In a tuning fork, the waves produced in the prongs is transverse whereas in the stem is longitudinal.
4. 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.
5. 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 length
and 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=0
y1 = 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 navt
where ,
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 Imin
i.e., 2π nAt = 0,π, 2π 2π nAt = π/2, 3π/2
i.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)

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

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
1. When source is in motion and observer at rest
1. when source moving towards observer
1. when source moving away from observer
Here V = velocity of sound
VS = velocity of source
ν0 = source frequency.
1. When source is at rest and observer in motion
1. when observer moving towards source
1. when observer moving away from source and
V0 = velocity of observer.
1. When source and observer both are in motion
1. If source and observer both move away from each other.
1. If source and observer both move towards each other.
1. 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
1. The motion of the listener causes change in number of waves received by the listener and this produces an apparent change in frequency.
2. The motion of the source of sound causes change in wavelength of the sound waves, which produces apparent change in frequency.
3. 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 superposition
y = y1 + y2 = (2A cos kx) sin ωt …(i)
y = AS sinωt
It 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
1. When a string vibrates in one segment, the sound produced is called fundamental note. The string is said to vibrate in fundamental mode.
2. The fundamental note is called first harmonic, and is given by, where v = speed of wave.
3. If the fundamental frequency be then , , … are respectively called second third, fourth … harmonics respectively.
4. If an instrument produces notes of frequencies …. where ….., then is called first overtone, is called second overtone, is called third overtone … so on.
5. Harmonics are the integral multiples of the fundamental frequency. If ν0 be the fundamental frequency, then nν0 is the frequency of nth harmonic.
6. Overtones are the notes of frequency higher than the fundamental frequency actually produced by the instrument.
7. 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 Correction
It 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 (l2l1)
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

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 Time
Sabine 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 = 0
At 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.
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