BS Physics (Foundation) Series

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Module # 45
Interference & Diffraction of Waves
Interference
Suppose two different sets of waves of the same amplitude
formed on the surface of water approach each other. The waves
may have originated from the opposite ends of a string or they
may be two circular waves formed on the surface of water when
two stones are dropped in it. When two sets of waves meet, they
are neither reflected nor absorbed by each other, but, one simply
passes over the other. However, at those points, where the waves
meet, the net amplitude of the combined wave will be the
algebraic sum of the displacements of two separate waves. By
interference, we mean the interaction of two waves passing
through the same region of space at the same time. It may be
observed that if, at a given point, the crests or the troughs of the
two waves arrive simultaneously, then, combined wave is larger
than either of the two waves. This is called constructive
interference. If, however, the crest of one wave arrives
simultaneously with the trough of the other wave, or, the trough of
one wave arrives simultaneously with the crest of the other wave,

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then, the two waves will cancel each other and no wave will be
observed. This is called destructive interference.
Light is a form of wave motion, giving out energy which is
distributed uniformly in the surrounding medium. If two sources of
light giving out continuous waves of the same amplitude are held
close to each other, then, the distribution of energy is not uniform
in the surrounding. At some places, the two waves reinforce (or
support), and at some other points, they cancel each other's
effect. This non-uniform distribution of light energy due to
superposition of two or more waves is called as interference.
That is, in other words, when two-phase coherent waves of the
same amplitude passing through a medium at the same time in
the same direction superimpose upon each other, then, they
reinforce (support) each other at some points and cancel each
other at the other points. As a result, we get dark and bright
fringes on the screen. The bright and dark regions or lines are
called interference fringes. Such a phenomenon is called
Interference.

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Conditions of Interference
(1) The sources should be monochromatic, that is, of single
wave length. A monochromatic source of light emits light of only
one wave length or one color.
(2) The sources should be coherent, that is, they should be
derived or split into two from the single source of light.
(3) The two sources should be close together.
(4) The principle of linear superposition should be applicable.
Types of Interference
There are two types of interference.
(1) Constructive Interference
(2) Destructive Interference
Constructive Interference
If the two light waves superimpose or combine so as to produce a
resultant wave of amplitude greater than that of either of the
individual waves, then, the interference is called constructive one.
OR
When two light waves meet at a point in such a way that they
reinforce each other's effect and brightness is observed, then this
type of interference is called as constructive interference.

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OR
If, at any point, the waves are in phase, that is, crest of one wave
falls upon the crest of the other or trough of one wave falls upon
the trough of the other, then, there is an increase in amplitude. It
is called constructive interference.
Constructive Interference (of Sound Waves)
In this case, two sound waves arrive at a point in the same phase.
The compression falls on compression and rarefaction falls on
rarefaction. As a result, the intensity of the resultant sound waves
increases which, in turn, increases the loudness of the sound.
Thus, in other words, when two sound waves of the same
frequency and amplitude pass through the same region of space
at the same time, then, at the place where compression of one
wave falls on the compression of the second wave, we get a
strong compression. Similarly, when rarefaction of one wave falls
on the rarefaction of the second wave, we get a strong rarefaction
and this results in (or gives rise to) a louder sound.
For constructive interference, the path difference between two
sound waves is given by:
S = 0,, 2, 3, ------------
S = n

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Where,
n = 0,1,2,3, ----- etc.
Condition of Constructive Interference
The constructive interference will take place if the path difference
between two waves is zero or integral multiple of wave length, i.e.
Path difference = d = 0,, 2, 3, --------
OR
In general
d = m where, m = 0, 1, 2, 3---------------- an integer
Destructive Interference
In this case, two sound waves arrive at a point in opposite phase.
The compression of one falls on the rarefaction of the other and
they cancel each other. As a result, intensity decreases which, in
turn, decreases the loudness of sound and we hear no sound or a
very faint sound.
OR
If the two light waves combine so as to produce a resultant wave
of amplitude less than the amplitude of the individual waves, then,
the interference is called destructive one.

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OR
When two light waves meet at a point so that they cancel each
other's effect and darkness (or dark fringe) is observed, then, this
type of interference is called as destructive interference.
OR
If the two waves are out of phase, that is, crest of one wave falls
upon the trough of the other wave, or the trough of one wave falls
upon the crest of the other wave, then, there is a decrease in the
resultant amplitude. This is called destructive interference.
Condition of Destructive Interference
The destructive interference will take place if the path difference
between two waves is odd integral multiple of half wave length.
That is;
Path difference = d = /2, 3/2, 5/2, -----------
OR
d = (m + ½) 
Where, m = 0,1,2,3 -------------
Reflection
We may hear the sound of a clap a second time if we are
standing near a cliff or in a large hall. The second sound of the

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clap is due to the bouncing back of the sound from the surface of
the cliff or the distant wall in the hall. The interval between
clapping and hearing the clapping sound depends upon the
distance between the person clapping and the reflecting surface.
Similarly, light is reflected by a mirror. This bouncing back of a
wave from a surface is called reflection. The angle at which the
wave is reflected is equal to the angle at which the wave is
incident on the surface. Waves coming from the source and
hitting an obstacle or barrier are called incident waves. The waves
that seem to originate from the barrier are called reflected waves.
These waves have the same frequency because they are
produced by the same source in the same medium having
uniform depth.
The turning back of the incident waves from the boundary of
another medium is called reflection of the waves.
The longitudinal waves such as sound waves obey the same laws
of reflection as are obeyed by transverse waves such as light
waves.
Reflection is the turning back of light when it travels from one
medium and falls on the surface of second medium.

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When light travelling through one medium, strikes the surface of
other medium, a part of it is turned back along a particular
direction in the same medium. This is called reflection of light.
Mirrors and highly polished opaque surfaces reflect light strongly.
Reflection of Light (& Formation of Image) in Spherical
Mirrors
Rays reflected from a concave mirror follow certain rules which
are necessary for the formation of images. These rules are
enlisted as follows.
Rule I:
A ray which is parallel to the principal axis after reflection from the
concave mirror passes through the principal focus.
Fig: Ray parallel to the principal axis
Rule II:
The ray passing through the principal focus after reflection from
the concave mirror becomes parallel to the principal axis.

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Fig: Ray passing through the principle focus
Rule III:
The ray which strikes the mirror after passing through the centre
of curvature retraces its path after reflection.
Fig: Ray passing through the center of curvature
Rule I V:
The ray that strikes the mirror at its pole is reflected back at an
angle of reflection which is equal to the angle of incidence.
Fig: Angle of incidence = Angle of reflection

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With the help of any two of the rays mentioned above we can
locate the position and describe the nature of the image of an
object placed in front of a concave mirror.
Laws of Reflection of Light
There are two laws of reflection of light which were discovered by
a Muslim scientist Ibn-ul-Haitham.
1 Angle of incidence is equal to the angle of reflection,
i.e. i = r.
2 The incident ray, the normal at the point of incidence and the
reflected ray all lie in the same plane.
A ray of light OA coming from an object is incident obliquely on
the reflecting surface of a plane mirror MM at a point "O". "O" is
called point of incidence and the ray OA is called the incident ray.
The incident ray is reflected from the mirror along OB, so the ray
OB is called the reflected ray. Draw a normal “NO” on the point O.
The angles AON and BON made by incident and reflected rays
with the normal NO are known as angle of incidence and angle of
reflection respectively. These angles are donated by i and r
respectively.
These angles are found to be the same in magnitude. Also, the
incident ray, the reflected ray and the normal at the point of

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incidence lie in the same plane.
Fig: Regular reflection showing the angle of incidence
and reflection.
Irregular Reflection of Light
When the parallel rays of light strike an irregular surface such as
a white paper or a painted wall, the angles of incidence at
different points of the surface are different due to which the rays
are reflected in different directions and light is scattered in
different directions. This random scattering is due to the highly
irregular nature of these surfaces, which can be observed by
using a microscope.
This is called irregular or diffused reflection of light.
As a consequence of the roughness of surfaces, the angle of
incidence does not remain the same for each ray and the
reflected rays, therefore, scatter in different directions.

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Fig: Irregular Reflection
Importance of Irregular Reflection
(1) Non luminous objects can be seen due to irregular reflection
of light.
(2) The dust particles in air scatter the sunlight in different
directions due to irregular reflection of light.
(3) With this type of reflection light reaches even those places
where sunlight cannot reach directly.
(4) It is such reflection due to which the sunlight reaches us
before sunrise and persists for sometimes even after the sunset.
Refraction of Light
When a ray of light enters from one medium into another
obliquely, a change in its velocity and direction takes place. This
change of velocity and direction as it enters from one medium into
another is called refraction of light.

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The angle between the incident ray and the normal is known as
the angle of incidence whereas the angle between the refracted
ray and the normal is called angle of refraction.
It has been determined experimentally that as light passes from a
rarer to a denser medium, it bends towards the normal. However,
a light ray passing from a denser to a rarer medium bends away
from the normal.
If the light passes normally (or perpendicularly) through the
surface separating two media, there is no change in the direction.
This is because when i is zero then r is also zero.
Refraction of Light through Water
Light entering from air into water bends towards normal and when
it emerges out of water, it bends away from normal.
Because of this reason, coin or pebble lying under water does not
appear at the same position where it actually lies and it looks
slightly raised. That is why, objects seem larger in water and
apparent depth of water pond seems less than real depth. So
refractive index of water is
Real Depth
Refractive index = -------------------------
Apparent Depth

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Laws of Refraction of Light
There are two laws of refraction of light.
(1) The incident ray, the refracted ray and the normal at the
point of incidence all lie in the same plane.
(2) The ratio of sine of angle of incidence i to the sine of
corresponding angle of refraction r is always constant for all rays
passing from one medium to another. This constant is called
refractive index denoted by n.
Sine of angle of incidence
Refractive index = --------------------------------------
Sine of angle of refraction
Sinei
n = ------------
Siner
This is also known as Snell’s law.
Diffraction of Light
The phenomenon of bending of light around the edges of an
opening or obstacle placed in its path is called diffraction of light.
OR
The spreading of light waves into the geometrical shadow of an
obstacle and the re-distribution of light intensity resulting in dark
and bright fringes is called diffraction of light.

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Explanation
One of the consequences of wave nature of light is that it spreads
as it travels, and when it encounters an obstacle, then, it bends
around it somewhat and passes into the region behind the
obstacle. To understand it, take the example of a person speaking
from one room to another through a hole in the wall. The person
in the other room may not necessarily stand in front of the hole to
hear the voice. Nevertheless, he can hear the voice regardless of
where he stands. Let us consider this phenomenon in case of
light.
When light shines on a slit placed in front of a screen as
shown in the figure below:
Fig: (a) Patch of light due to a circular slit

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Fig: (b) the dark and bright fringes in the geometrical shadow of a
circular aperture
then, we expect a sharp patch of light and a shadow. A close
observation of the light distribution on the screen reveals dark and
bright fringes near the edges, as shown in figure (b). Similarly, we
find that whenever an obstacle is placed in the path of light, then,
instead of sharp shadow, we get dark and bright fringes near the
edges. This can be explained by assuming that light waves bend
(or spread out) around obstacle or sharp edges of an aperture.
Diffraction of light can take place only if the size of the obstacle is
so small that it may be comparable to the wave length of the
incident light. For example, a slit, a wire, a small bore or a straight
edge can produce diffraction effects under proper conditions.
This phenomenon is exhibited when light passes through a
narrow slit or aperture. Actually, there is no significant difference
between interference and diffraction phenomena. Interference is

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usually referred to as the superposition of only a few secondary
wavelets originating from the wave front whereas diffraction is
concerned with the superposition of a very large number of
secondary waves. The visual difference between the interference
fringe pattern and diffraction fringe pattern is that interference
fringes are equally spaced whereas the diffraction fringes are
wide near the obstacle and go on becoming narrower as they
move away into the shadow region.
The phenomenon of diffraction supports that the wavelength of
light waves is much shorter than those of sound because sound
can diffract about large obstacle whereas light cannot.
Types of Diffraction
There are two types of diffraction phenomenon known as
Fresnel's diffraction and Fraunhofer diffraction.
Difference between Interference and Diffraction
Sr.
No.
Interference Diffraction
1 Interference is the result of
superposition of only a few
secondary wavelets from
two wave fronts originating
Diffraction is the result of
superposition of a very large
number of secondary wave-
lets coming from different

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from the same source. That
is, in case of interference,
two coherent waves
superpose at a point to give
constructive or destructive
interference.
parts of the same wave
front. That is, in case of
diffraction, secondary
wavelets from the various
parts of the same wave front
interfere constructively or
destructively.
2 Interference fringes are
equally spaced. They can
be of the same width.
Diffraction fringes are not of
the same width and not
equally spaced. They are
wide near the obstacle and
become narrower moving
away towards the shadow
region.
3 All bright fringes are of the
uniform intensity.
All bright fringes are not of
the same intensity.
4 Points of minimum intensity
are perfectly dark.
Points of minimum intensity
are not perfectly dark.
5 All bright bands are of same
intensity.
All bright bands are not of
the same intensity.
6 Interference can take place There can be no diffraction

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without diffraction. without interference.
7 Interference is the name
given to the resultant effect
of the superposition of
waves.
Diffraction means the
bending or the change in the
path of light when it passes
across obstacles.
Common Factor between Interference & Diffraction
Superposition of waves is common to both interference and
diffraction.
Grating
It is a transparent glass plate, having a number of parallel and
equidistant lines ruled upon it.
Diffraction Grating
An arrangement consisting of large number of parallel slits of the
same width and separated by equal opaque spaces is known as
diffraction grating. It is based upon diffraction principle and is
used for the determination of wavelength of light using a
spectrometer. It can also be used to disperse a spectrum into
component wavelengths just as a prism disperses white light into
component colors (wavelengths).

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Fraunhofer used the first grating which consisted of a larger
number of parallel wires placed very closely side by side at
regular intervals. Now, gratings are constructed by ruling
equidistant parallel lines on a transparent material such as glass,
with a fine diamond point. The ruled lines are opaque to light
while the space between any two lines is transparent to light and
acts as slit. This is known as plane transmission grating. On the
other hand, if the lines are drawn on a silvered surface (plane or
concave) then, the light is reflected from the positions of mirrors in
between any two lines and it forms a plane or concave reflection
grating. When the spacing between the lines is of the order of the
wavelength of light, then, an appreciable deviation of the light is
produced.
Some gratings may have as many as 12,000 lines per centimeter
to produce a diffraction of visible light. The spacing d between the
centers of adjacent lines is called the grating element and is given
by d = 1/N, where N is the number of lines in one unit length of a
diffraction grating.
Diffraction Grating Equation
A parallel beam of light is made to fall normally (perpendicularly)
on the grating. The beam is brought to focus at the point O by
means of a convex lens, where there will be bright line. If the

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diffracted rays are focused at the point ‘P’ on the screen, as
shown in the figure below, then, the path difference between the
rays 1 and 2 is ab.
But ab/d = sin or ab = d sin
Fig: Schematic Diagram of Diffraction by a Diffraction Grating
The point P will be bright if the path difference is 0, , 2, 3,-----.
Thus, d sin = m, where m = 0, 1, 2, 3, ------ is called the order of
the spectrum. (The spectrometer is used to measure m and  for
this equation.) When m = 0, 1 or 2, then, it is called the central
maxima, first order maxima or second order maxima respectively.
The above equation is called Diffraction Grating Equation.
Fraunhofer Diffraction
In this type of diffraction, the source and the screen are at infinite
distance from the diffracting aperture. Thus, the light rays before

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falling and after leaving the diffracting aperture are parallel and
corresponding wave fronts are plane.
The convex lenses are used to focus the diffracted rays.
Fresnel's Diffraction
In this type of diffraction, the source of light and the screen on
which the diffraction is to be observed are at finite distance from
the diffracting object.
Diffraction of X-Rays by Crystals
The diffraction can take place only when spacing of slit is
comparable to wavelength of the light wave. That is, in other
words, diffraction effects can be observed only if the spacing
between the lines ruled on the grating is of the order of magnitude
of the wavelength of light used. X-rays are electromagnetic waves
of very short wavelength (of the order of 10-10
m = 1 Angstrom).
As we cannot get a slit of this spacing, therefore, ordinary grating
cannot be used for diffraction of X-rays. However, a salt crystal
can be used for this purpose. In a crystal, the atoms are regularly
arranged in various planes.
The diffraction patterns that one observes are rather complicated
because of the three dimensional nature of the crystal.
Nevertheless, x-ray diffraction has proved to be an invaluable

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technique for determining crystalline structures and understanding
structure of matter.
Photographic film
Fig: Diffraction of x-rays by crystals
The above figure is an experimental arrangement for observing x-
ray diffraction from a crystal, such as one of sodium chloride. The
diffracted beams are very intense in certain directions,
corresponding to constructive interference from waves reflected
from layers of atoms in the crystal. The diffracted beams can be
detected by a photographic film where they form an array of spots
known as a "Laue Pattern'. The crystalline structure is determined
by analyzing the positions and intensities of various spots in the
pattern.