1. Wave mechanics was introduced by De Broglie in 1924 and is based on the idea that particles can be regarded as waves described by the Schrodinger wave equation. 2. Waves transfer energy but not matter. Different types of waves include mechanical, electromagnetic, and matter waves. Mechanical waves require a medium while electromagnetic waves do not. 3. The wavelength is the distance between two consecutive peaks or troughs of a wave. The velocity, wavelength, and frequency of a wave are related by the equation v = λf.

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Module # 42
Waves, Wavelength & Wave Front
Waves
Wave Mechanics
Wave mechanics was introduced by De Broglie in 1924.
It is one of the forms of quantum mechanics that developed from
the De Broglie's theory that a particle can also be regarded as a
wave. Wave mechanics is based on the Schrodinger wave
equation describing the wave properties of matter.
Wave Motion
A mechanism by which energy is transferred from one place to
another is known as wave motion.
The waves carry energy, but, there is not any transfer of matter in
case of waves. When a drummer beats a drum, then, its sound is
heard at far distant points. The sound carries energy as it has the
ability to move the diaphragm of the ear. When a stone is dropped
in the still water in a pond, then, water waves move steadily along
the water until they reach the shore. If there is a small floating
object like a piece of cork in the way of these waves, then, it will
move up and down near its own location, which indicates that the
molecules of water do not move along with the wave.

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Waves are responsible for the enormous amount of heat and light
received from the sun and other sources. When a bulb is turned
on, the room is flooded with light. Light waves also carry energy.
Radio and television programs are carried to us by
electromagnetic waves. It is possible to transmit an electric signal
(or a message) from one place to another due to waves.
Although these various processes of transport of energy are
different, yet they have a common feature which we will call wave
motion.
Transmission of Waves
Whenever a wave, moving in certain medium, comes across
another medium, then, only a portion of it is reflected back and
the remaining portion passes into the other medium. This
passage of the wave from one medium into the other is known as
transmission of the wave. The relative portions of wave those are
reflected and transmitted depend upon the values of inertia and
elasticity of one medium as compared to the other.
Wave Process
There are essentially two ways of transporting energy from the
place where it is produced to that place where it is intended to be
utilized. The first involves the actual transport of matter. For

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example, a bullet fired from a gun carries its kinetic energy with it
which can be used at another location. The second method by
which energy can be transported without transfer of matter is
much more useful and important and it involves what we call a
wave process.
Types of Waves
There are three different types of waves:
(a) Mechanical Waves
(b) Electromagnetic Waves
(c) Matter Waves
Mechanical Waves
The waves which require a medium for their propagation are
known as mechanical waves. For example, sound waves, waves
on the surface of water and waves along a string.
Matter Waves
The waves associated with the moving particles are called matter
waves, for example, moving elementary particles such as
electrons moving with very high velocities show wave as well as
particle nature. That is, in some cases, they move like a wave and
in others like a particle.

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Stationary or Standing Waves
A stationary wave is formed when two equal progressive waves
are superposed on one another while travelling in opposite
directions.
OR
The interference of two waves having the same amplitude and
time period but travelling in opposite directions in a straight line
gives rise to stationary or standing waves.
OR
If two waves of the same amplitude and frequency, travelling in
the opposite directions, meet one another, the resulting
interference pattern gives rise to what are called standing waves
or stationary waves.
Stationary waves may be produced by means of a single source.
For example, the incident waves may interfere with the reflected
waves to produce stationary waves. As the incident waves and
the reflected waves originate from the same source, pass through
the same medium, therefore, they possess the same amplitude
and frequency. Such a wave is called a stationary wave because
it does not appear to be moving. The points of destructive
interference called nodes and of constructive interference called

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antinodes, remain in fixed positions. Stationary waves are formed
at more than one frequency. The distance between a node and an
antinode is one quarter of the wavelength.
Thus, a standing (stationary) wave is produced when two waves
of the same amplitude and frequency that are travelling in the
opposite directions are combined.
Characteristics of Standing or Stationary Waves
(1) No energy is transferred from particle to particle in a
stationary wave.
(2) All particles, except nodes, perform S.H.M. with the same
period as the component waves.
(3) At nodes, strain is maximum and the amplitude is zero. But,
at antinodes, strain is minimum and the amplitude is maximum.
(4) Distance between two consecutive nodes or antinodes is
equal to half of the wave-length (/2).
(5) Distance between a node and neighboring antinode is /4.
(6) Between two consecutive nodes, there must be an antinode
and between two consecutive antipodes, there must be a node.

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Compressional Stationary Waves
Consider a spring of length ℓ stretched between two fixed clamps
as shown in the figure below.
compress here
Fig: The compressional stationary waves are set up in the spring
by compressing it at its mid-point.
If the spring is compressed at its centre and then released, two
compressional waves will originate from the center and will move
towards the two ends. When these waves reach the two ends,
they are reflected back, thus forming stationary waves. The
frequency f1 of the waves is such that nodes are formed at the
ends, where the spring has no motion and an antinode is formed
at the centre.
Now, the central portion of the spring vibrates parallel to length of
the spring with the maximum amplitude. If ν is the speed of the
waves, then, as in case of stretched string, the frequency f1 = v/2ℓ.
The other quantized frequencies will be
f2 = 2f1, f3 = 3f1, and fn = nf1

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where, n is a positive integer. The various frequencies are
generated depending upon the position of the initial compression.
Waves as Carriers of Energy
It has been observed experimentally that waves transmit energy.
We know that a piece of cork placed on the surface of water in a
ripple tank moves up and down as waves pass over it. This shows
that energy has been imparted to the cork by the waves.
Fig: Waves carrying energy reaching a certain cross sectional
area
It can easily be shown that a wave with a large amplitude
possesses more energy than a wave with a smaller amplitude.
How much energy is possessed by waves? We can deduce the
result by considering waves reaching a certain cross sectional
area of a layer of a medium in phase. The layer is held
perpendicular to the waves. The waves transfer their energy to
the layer which begins to vibrate with simple harmonic motion.
This arrangement may lead us to the conclusion that the energy
transported through a unit area held perpendicular to the wave in

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one second is 1/2 c2r2 where c is the velocity of waves,  is the
density of the medium in kgm-3
, and r is the amplitude of the
wave. This relation shows that the energy transported by a wave
is proportional to the square of the amplitude of the wave as c, 
&  in the above relation are constants.
Periodic Waves
If water in the tray is disturbed periodically after equal intervals of
time, then, waves produced one after the other pass through a
point in the medium. Such waves are called transverse periodic
waves. Most of the waves that we usually come across in the
actual physical world are periodic waves, so their detailed study is
very essential.
Straight Periodic Waves
Straight periodic waves can be produced in a ripple tank by
dipping a straight rod periodically into the water. This can be done
by means of a mechanical arrangement driven by a small electric
motor. The rate of dipping can be changed by changing the speed
of the motor. The waves produced in a ripple tank are always
transverse in nature. The pattern of transverse periodic waves as
produced on a photographic screen is shown in the figure below:

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Fig: Instantaneous photograph of moving periodic straight waves
striking ripple tank.
The bright bands correspond to crests. Thus, dark and bright
bands are seen on the screen. The distance between the centers
of two consecutive bright bands is called wave length ''.
In order to observe the motion of water (medium), we put a small
piece of paper on the surface of water and mark its shadow on
the screen. Now, we produce the straight periodic waves on the
surface of water. As the wave passes by, paper continues to
move up and down. It means the particles of water also oscillate
up and down. Thus, the paper performs a simple harmonic motion
in a direction at right angles to the direction in which the waves move.
Water Waves
The surface waves that propagate across deep water are similar
to the waves on springs. One marked difference between the two
is that water particles do not execute a strict transverse motion
but undergo a circular motion at their respective positions.

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Water Wave Wave direction Wave Crust
Wave Trough
Fig: Water molecules move in circular orbits when wave passes
by
Water molecules move around circular paths at their respective
positions when the wave advances. At the positions known as
crests, the molecules move in the direction in which the wave
advances, while, at positions known as the troughs, the molecules
move in the opposite direction. There is no net transfer of mass
from one position to the other as the wave moves.
Complex Wave
The superposition of any number of sine waves having the same
frequency gives rise to a resultant wave which is again a sine
wave. If we super-impose waves that have different frequencies,
the resultant wave is called a complex wave.
Infrared Waves
Infrared waves are also called heat waves. Infrared waves are
radiated by hot bodies at different temperatures. The earth's
atmosphere is at mean temperature of 250 K and radiates

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infrared waves with a wavelength having a mean value of 10
micrometers or 10-5
m (1m = 10-6
m).
Visible Waves
Visible waves have a wavelength range between 400 and 700
nanometers. The Peak of the solar radiation is at a wavelength of
about 500 nm. The human eye is most sensitive to this
wavelength.
Radio Waves
Radio waves are electromagnetic waves with a large range of
wavelengths from a few millimeters to several meters.
Microwaves
Microwaves are radio waves of shorter wavelengths between
1m and 300m. Microwaves are used in radar and microwave
ovens.
Ocean Waves
Surface waves on open water build up due to the action of winds.
But, anyone who has ever seen the sea, knows, that the waves
are not the simple ideal waves as those observed in the ripple
tank. As the winds are constantly changing their direction and
strength, the ocean waves are, therefore, complex and irregular

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which may range from ripples to giant crashing waves. In
describing these waves, we can only refer to average properties.
The size of the ocean waves which are wind-driven based
depends upon the speed and the duration of the wind. In a strong
and steady wind of about 50 km h-1
, the waves will continue to
grow for a day, eventually, reaching an average height of about
4m. For gale winds travelling at 90 km h-1
, the average height of
waves is about 13 meters or 45 feet.
Ultrasonic Waves
The sound waves above the audible range of frequencies are
called ultrasonic waves.
Ultraviolet Waves
The wave length of ultra violet wave ranges from 380nm down to
60nm. These are emitted by hotter stars having a mean
temperature greater than 25000°C.
Ripple Tank
The apparatus used to study the various features of the wave
phenomenon is called ripple tank as shown in the figure below.

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Screen
Fig: The image of the wave is seen on the screen which is just a
piece of white drawing paper placed under the ripple tank.
Construction
It consists of a rectangular tray containing water, fitted with a
glass bottom (i.e. transparent). The tray is mounted on four legs
and a screen is placed well below the glass bottom. A lamp is
fixed above it and switched on while taking the photograph.
Working
The water waves which appear on the surface of water can be
projected on a screen placed below the glass bottom. The
transparent glass bottom makes it possible to project image of
waves formed on the surface of water in the tray on a screen
placed well below the glass bottom. The image is projected
because crests of water act as converging lenses and tend to

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focus light from the lamp while the troughs act as diverging lenses
and tend to spread it. As a result, we get alternate bright and dark
bands on the screen. We should note here that this happens only
when the bulb is switched on.
Straight Pulse
A straight pulse can be produced by dipping straight rod into the
water of tray.
Circular Pulse
A single circular pulse can be produced by dipping a finger into
the water. The crest will appear as bright circular band on the
screen moving radially out.
Coherent Sources
Two sources of waves are said to be coherent if there is a fixed
phase relationship between the waves they emit during the time
the waves are being observed. It does not matter whether the
waves are exactly in step when they leave the sources, or exactly
out of step, or anything in between; the important thing is that the
phase relationship stays the same.
Production of Coherent Sources
A common method for producing two coherent light sources is to
use one monochromatic source to illuminate a screen containing

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two small openings (usually in the shape of slits). The light
emerging from the two slits is coherent because a single source
produces the original light beam and the two slits serve only to
separate the original beam into two parts. A random change in the
light emitted by the source will occur in the two separate beams at
the same time, and interference effects can be observed.
Incoherent Sources
If the sources shift back and forth in relative phase while the
observation is made, then such sources are called incoherent
sources. There will be no interference for such sources.
Fundamental Wave and Harmonics
Complex waveforms are formed due to superposition of
sinusoidal waves of different frequencies. They consist of a
fundamental wave and a number of other sinusoidal waves. A
wave which has lowest frequency is called fundamental wave. It is
represented by f. A number of other sinusoidal waves whose
frequencies are integral multiples of the fundamental or other
frequencies such as 2f, 3f, 4f, 5f etc. are called overtones or
harmonics.
The term harmonic means the frequencies which are an exact
integral multiple for the lowest or fundamental frequency.

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A fundamental wave is called the 1st harmonic. The second
harmonic has frequency twice that of fundamental. The third
harmonic frequency is thrice of the fundamental frequency and so
on.
Wavelength
The wavelength λ (Lambda) is defined as the distance between
two successive particles which are at exactly the same point in
their paths and are moving in the same direction.
Alternatively, the distance traveled by the wave in one cycle is
called wavelength. The wave length is denoted by a Greek letter λ
(lambda).
Wavelength of light can be measured using Michelson
Interferometer which is based on the principle of interference.
OR
The distance between similar positions of two consecutive crests
or troughs is known as wave length. It may also be defined as the
distance between similar positions of two consecutive
compressions or rarefactions. It is denoted by  (lambda).
Equation between Velocity, Wavelength and Frequency
Suppose the velocity of a wave is v, its wavelength is  and its
frequency is f. Let T be the time period. When the particle of the

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medium completes one vibration, the distance travelled by a wave
is its wavelength.
As the frequency is defined as the number of vibrations
completed by a vibrating body in one second, so time for one
vibration is reciprocal of frequency.
Time period = 1/frequency
OR
T = 1/f
OR
f = 1/T
As distance covered in T seconds = 
So, distance covered in one second = /T
But,
Distance covered in one second = Velocity
So,
Velocity = /T
OR
v = /T =  x f

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Wave Front
Any line or section taken through an advancing wave in which all
the particles are in the same phase is called the Wave Front.
OR
Whenever, a wave passes through a medium, then, its particles
execute SHM. The path (locus) of all the particles of the medium
having the same phase and same state of vibration is known as
Wave Front.
Explanation
A source of light emits electromagnetic waves which propagate in
space with different points of vibrations having different phases
and the wave front is the locus of all the points in the same phase
of vibration. Thus, in the case of electromagnetic waves, a sphere
with its centre at the source will be a wave front with all the
vibrations in phase.
Similarly, in the case of water waves, any circle drawn with its
centre at the source is a wave front. All the points on the wave
front will not only be in the same phase but will also have the
same displacement.
If we dip one end of a stick into water, then, circular waves are
produced as shown below.

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The circles as shown in the above figure are the crests and,
therefore, represent the wave fronts.
Crest
The portion of water above the mean level is said to form a crest.
Trough
The portion below the mean level is called trough.
OR
In a transverse wave, the part of the medium which is below the
mean level is called a trough.
Plane Wave Front
At a very large distance i.e. infinity from the source, a small
portion of a spherical wave front will become very nearly plane.
This type of wave front is called as plane wave front as shown in
figure below.

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Fig: Plane Wave Front
Spherical Wave Front
In case of a point source of light in a homogeneous medium, the
wave fronts will be concentric spheres with centre at the source S.
Such a wave front is known as spherical wave front.
Fig: Spherical Wave front
Plane Wave Front
Large
Distance

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Ray of Light
A ray of light gives the direction of propagation of light and is
always along the normal to the wave front. Hence, a plane wave
front represents parallel pencil of rays while a spherical wave front
represents a diverging pencil of rays.
Pencil of Rays
A collection of parallel rays is called a pencil of rays.
Wave Intensity
Wave intensity is defined as the power transmitted per unit area
of the wave front.
Resonance
Phenomenon in which there is a remarkable increase in the
amplitude of a body when the period of the force applied to it is
equal to its natural time period is called resonance.
OR
When a periodic force having a time period equal to the time
period of a vibrating body is applied on that body, then, the
amplitude of the vibration is increased. This process is called
resonance.

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Explanation
1 Consider the motion of a simple pendulum. When this
pendulum is displaced from its equilibrium position, then, it starts
vibrating with a certain time period, T. Its frequency and time
period can be calculated as follows:
__ ___
T = 2π ℓ/g & f = 1/ T = 1/2π  g/ℓ
Where ℓ = Length of the pendulum and g = Acceleration due to
gravity
Thus, period and the frequency depend only upon the two factors,
i.e.
(1) Length of the simple pendulum, and
(2) Acceleration due to gravity.
On disturbing, the pendulum vibrates with the same time period.
Its time period is known as natural time period and its frequency is
natural frequency. These two can be changed by changing the
length of the pendulum.
2 Consider a long string or a wire stretched tightly between
two pegs. Four pendulums, A, B, C, and D of different lengths are
tied to the string (or wire). Another pendulum E of the same length
as that of B is also tied to the same string or wire as shown below:

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Fig: Resonance
When pendulum E is set swinging, then, all the pendulums begin
to vibrate, but, the pendulum B begins to vibrate with increasingly
larger amplitude.
As pendulum E is set into vibration, it transfers its motion to the
string or wire. This string (or wire), in turn, imparts the same
motion to the pendulums tied to it. Since frequency of a pendulum
depends upon its length, therefore, the pendulum B gives positive
response because its natural frequency agrees (coincides) with
the frequency of the motion imparted by E to the string. All other
pendulums having frequencies different from that of B do not
respond to the same extent to the motion imparted from the
string. The phenomenon under which B begins to vibrate is known
as resonance.
B

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Resonance is the response of an object to a vibration or a
periodic force acting on it. This response is the greatest when the
periodic force has the same period as the object's natural period.
3 Consider a pendulum hanging freely from a support. If we
slightly disturb it, then, it will begin to oscillate. The time period of
these oscillations (called natural oscillations) depends upon the
length of the pendulum. If we hold the bob in our hand and move
it to and fro, then, the time period will depend upon the hand. If
we move the hand rapidly, then, the time period will be small and
it will be large if we move the hand slowly. These vibrations are
called forced vibrations. Now, if the frequency of the forced
vibrations coincides with the natural frequency of the pendulum,
then, the amplitude of the pendulum increases.
4 A simple apparatus for demonstrating resonance in sound
waves consists of a vertical open tube partially dipped in water
contained in a beaker. A vibrating tuning fork is held slightly
above the upper end of the tube. The length of the air column is
adjusted vertically by moving the tube in or out of the water. The
sound waves generated by the tuning fork are reinforced when
the length of the air column corresponds to one of the frequencies
with which air column can vibrate. This arrangement can be used
to determine the velocity of sound in air.

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Whenever, a sound wave comes across a barrier, it is reflected
back in the same medium. In this process, the reflected waves
interact with the incident waves and produce stationary waves,
giving rise to a louder sound. The loud sound indicates that the
reflected waves are in resonance with the incident waves
produced by the vibrating tuning fork. Stationary waves consist of
alternate nodes and antinodes.
Measure the length of air column from water to the top of the
tube. The reflection of sound waves at the upper end takes place
a little distance above the open end. But, this is usually ignored,
unless, high accuracy is required. The speed of sound can be
experimentally calculated by using relation.
v =  x f
Where  is the wave length and is four times the distance (length
of air column) at which maximum loudness was obtained, f is the
frequency of air column and is equal to the frequency of the
tuning fork.
Applications of Resonance regarding Frequency
Determination
(1) Resonance can be used to determine the frequency of a
given body. A second body, the natural frequency of which is

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known, is made to act on the given body. If it produces
resonance, then, it is concluded that the given body has the same
frequency as the second body.
(2) It is used to find the natural frequencies of the different
bodies.
Examples of Resonance
(1) Sometimes, a part of the car begins to vibrate very violently
at a certain speed of the engine or the car. If the speed of the car
is changed from that value, then, the vibrations cease.
(2) Soldiers are instructed to march out of the step while
crossing a certain bridge because if the frequency of their steps
coincides with the frequency of the bridge, then, a vibration of
dangerously large amplitude may be produced and the bridge
may collapse.
(3) Many cities with tall buildings have already refused to allow
supersonics to fly over them to avoid resonance in buildings.
(4) In the swing, if pushes are given at the correct interval
(moments), which coincide with the period of the swing, then, the
amplitude of the swing can be made quite large. The children
playing with the swing apply this device often.

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Similarly, a feeble puff of air, blown at regular intervals of time,
increases the amplitude of vibration of the pendulum to a large
extent.
(5) Tuning a radio is an example of electrical resonance. By
tuning a dial, the natural frequency of an alternating current in the
receiving circuit is made equal to the frequency of the waves
broadcast by the desired station.