1. No category

AS Physics Unit 5 - Waves
1
WHAT IS WAVE MOTION?
The wave motion is a means of transferring energy from one point to another without the transfer
of any matter between the points. Waves may be classified as mechanical waves and
electromagnetic waves.
Mechanical Waves:
 Mechanical waves (example water waves, sound waves, waves in stretched springs) require
presence of material medium or some matter for their propagation (travel).
 When it travels from point A to point B, it is because of the disturbance of some kind at A
has caused the particles of the medium being displaced. This particle drags its neighboring
particle with it, so that it too becomes displaced and has a similar effect on next particle, and
so on until the disturbance reaches at point B.
 There are two types of mechanical waves, that is, transverse waves and longitudinal waves.
o
The transverse wave is the one in which the disturbance occurs perpendicular
to the direction of travel of the wave.
o
The longitudinal wave is one in which the disturbance occurs parallel to the
line of travel of the wave.
Note 1: The transverse and longitudinal waves are also called periodic waves or
progressive waves because they consist of cycles or patterns that are produced over and
over again by the source.
Note 2: A water wave is neither transverse nor longitudinal, since water particles at the
surface move clockwise on nearly circular paths as the wave move from left to right.
Electromagnetic waves:
1. Electromagnetic (EM) waves (example light
waves, radio waves, x-rays etc) can travel
through vacuum; they do not require any
medium (matter) to travel through and their
travel is reduced, to some extent, by the
presence of any medium. The common
examples of medium are glass, water, air
etc.
2. EM waves are transverse in nature and
comprise of oscillating electric (E) and
magnetic (H) fields perpendicular to the
direction of wave.
Prepared by Faisal Jaffer, revised on 7-Mar-10
3. All electromagnetic waves including light waves travel with the same speed that is 3×108 m/s.
They all have different frequencies and wavelengths. The band of different electromagnetic
waves called electromagnetic spectrum is shown in the figure below.
AS Physics Unit 5 - Waves
3
Important wave definitions:
Wavelength (λ): The wavelength ‘λ’ of a wave, represented by the Greek letter λ (lambda). It is
the distance between the two successive crests or troughs.
Frequency (f): The frequency ‘f’ is the number of complete waves generated per second. The unit
of frequency is ‘vibration per second’ or ‘hertz (Hz)’.
Period (T): It is the time taken for a wave to travel one complete cycle or oscillation. Unit is
seconds.
Speed of wave motion (v): The speed, ‘v’ of the wave
is the distance moved in the direction of travel of the
wave by a crest or any point on the wave in 1 second.
Displacement(s): The displacement, ‘s’ of the
particles of the medium is their distance from the rest
or central position in either direction.
Amplitude (a): The amplitude ‘a’ is the height of a
crest or the depth of trough measured from the central
position. It is the maximum displacement from the rest
or central position in either direction.
Wavefronts: The wavefronts are represented by
straight lines and can be thought as the crests or
troughs of the waves. They are at the right angle to the
direction of waves. The distance between the lines is
called wavelength.
Progressive wave: A progressive or traveling wave is the continuous disturbance of medium
which carries energy from the source to another point.
Phase difference: When the crests of two waves of equal wavelength are together the waves are
said to be in phase that is the phase difference is zero. If a crest and a trough are together, the
waves are completely out of phase that is they have a phase difference of π radians or 180o.1
Coherent Waves: Waves originate from same source, have same wavelength and have constant
phase difference between them is called Coherent waves.
Intensity of wave: Waves carries energy that can be used to do work. In case of sound waves the
work done in forcing the eardrum to vibrate. The amount of energy transported per second is
called power of the waves and is measured in joules per second (J/s) or watts (W). The intensity I
of sound wave is defined as the sound power P that passes perpendicularly through a surface
divided by the area A of that surface.
I = P/A
1
Exercise on page no 425: Muncaster
Prepared by Faisal Jaffer, revised on 7-Mar-10
The unit of wave intensity is power per unit area, or W/m2. In terms of amplitude of the wave
Intensity ⋉ (amplitude) 2
Thus if the amplitude of the sound wave is doubled, it carries four times more energy per second
away from the source.
Graphical representation of longitudinal waves and transverse:
In Longitudinal waves the motion of the individual
particles of the medium is in direction which is
parallel to the direction of wave or energy. The back
and forth moment of the vibrating object makes the
particles of the medium, for example air, move back
and forth. These back and forth vibrations are passed
on to adjacent neighbors by particle interaction; thus,
other surrounding particles begin to move rightward
and leftward, thus sending a wave to the right. Since
air molecules (the particles of the medium) are
moving in a direction which is parallel to the
direction which the wave moves, the sound wave is
referred to as a longitudinal wave. The result of such
longitudinal vibrations is the creation of compressions
(C) and rarefactions (R) within the air.
A transverse wave is a wave in which the particles of
the medium are displaced in a direction perpendicular
to the direction of wave or energy. The snapshot of the
transverse wave could be represented by diagram
The crest of a wave is the point on the medium which
exhibits the maximum amount of positive or upwards displacement from the rest position. Points
C and J on the diagram represent the troughs of this wave. The trough of a wave is the point on
the medium which exhibits the maximum amount of negative or downwards displacement from
the rest position.
Derivation of wave equation v = f λ:
Whenever the source of wave motion does
one cycle the wave moves forward by the
distance (x) of one wavelength (λ). Since f is
the number of cycles in each second, the
wave moved forward by fλ distance in this
time, and therefore the velocity (v) of the
wave is given by
x

x = λ (when x is the distance moved
equal to one wavelength)
λ
but x = vt (where v is the velocity of the wave and t is the time)
vt = λ
or
v = λ/t
‘t’ here is the time for one oscillation therefore it can be written as ‘T’ the period of wave.

v = λ/T
But the f = 1/T (where ‘f’ is the frequency of the wave)

v = f λ or
=
where c is the speed of light waves in vacuum which is 3×108m/s.
AS Physics Unit 5 - Waves
5
POLARIZATION OF LIGHT WAVES:
Light wave is an electromagnetic wave which travels through
vacuum. Light waves are produced by vibrating electric charges.
It is a transverse wave which has both an electric and a magnetic
field component. The electric and magnetic vibrations of an
electromagnetic wave occur in numerous planes. A light wave
which is vibrating in more than one plane is referred to as
unpolarized light. Light emitted by the sun, by a lamp in the
classroom or by a candle flame is unpolarized.
It is possible to transform unpolarized light into polarized light.
Polarized light waves are light waves in which the vibrations
occur in a single plane. The process of transforming unpolarized
light into polarized light is known as polarization.
http://micro.magnet.fsu.edu/optics/lightandcolor/polarization.html
Demonstration of polarization using string and fence:
In the example, the vibrations in the rope will pass through a
narrow gap provided the longest axis of the gap is parallel to the
direction of the vibrations. Thus if the rope is vibrating up and
down, the motion will be transmitted through a vertical gap.
However, if the rope also passes through a horizontal gap the
vibrations can't pass through this.
Light can be polarized by transmission, reflection, refraction,
scattering. For AS course we are only going to study the
polarization by reflection method.
Polarization by reflection
If an unpolarized beam of light is incident on a glass surface at an
angle of about 57o , the light that is reflected from the surface is
completely plane-polarized. This can be checked by using
Polaroid film. At angles of incident other than 57o , the reflected
light is partially plane-polarized.
When the beam of light incident on glass at 57o and partially
reflected and partially refracted light make an angle of 90o between
each other, this is a particular incident angle at which light is
completely plane-polarized. All other angles the light is partially
plane-polarized.
Let’s consider the incident angle θB, refracted through an angle θ2
and refractive index of glass is ‘n’ than we can write
=
sin
sin
ℎ
ℎ
⇒
=
sin
sin(90 −
sin(90 −
⇒
=
sin
cos
= 90 −
)
) = cos
= tan
The equation = tan is called Brewster’s Law. We can find that light incident at the brewster
angle has reflected and refracted rays which are perpendicular to each other. If we replace the
value of n = 1.54 as refractive index of glass then the above equation gives θB = 57o.
Prepared by Faisal Jaffer, revised on 7-Mar-10
DIFFRATION OF WAVES:
When straight waves are incident on a small opening
formed between two vertical bars placed in the path of
waves, then the wavefronts emerge with a circular shape
and waves spread out in all directions from the opening.
The spreading of waves at the edges of obstacles is
called diffraction. The extent of diffraction is depend
upon the width of the opening and the wavelength of the
wave. It is more noticeable if the width of the opening is
almost equal to the wavelength of the wave.
INTERFERENCE OF LIGHT:
A set of two waves (such as light or sound) can combine
with each other to produce a resultant wave. The way in
which this combined wave is produced is called
interference or superposition of waves.
Constructive interference
Destructive Interference
The principle of superposition of waves:
“The principle of superposition states that whenever two waves are traveling in the same
region the total displacement at any point is equal to the sum of their individual
displacement at that point”.
When two identical waves (i.e., waves of same wavelength λ and amplitude) arrive at the same
point in phase – that is, crest-to-crest and trough-to-trough then according to the principal of
superposition, the waves reinforce each other and constructive interference happened. The
resulting total waves at a point has an amplitude that is twice the amplitude of either of the
individual waves, and in case of light waves, the brightness is greater than due to either wave
alone.
When two waves l1 and l2 start out in phase and are in phase when they reach at point P then
l2 - l1 is equal to the λ and the interference will be constructive
When l1 is the distance covered by wave 1 from the source
l2 is the distance covered by wave 2 from the source
λ is the wavelength of the wave
In general, when the waves start out in phase, constructive interference will result whenever the
distance travel by two waves are the same or differ by any number of complete wavelengths
l2 – l1 = nλ
where n = 0, 1, 2, 3 ….
Eq. 
When two identical waves arrive at any point out of phase with one another, or crest-to-trough
then the waves cancel each other and dark region appear on the screen. According to the principal
of superposition it is called destructive interference. That is
l2 - l 1 = ½ λ
AS Physics Unit 5 - Waves
or
l2 - l1 = (n + ½) λ
7
where n = 0, 1, 2, 3…
Eq. 
If the constructive and destructive interference is to continue occurring at a point, the sources of
the wave must be coherent. Two sources are coherent if the waves they maintain a constant phase
relation. Lasers are coherent sources of light, where incandescent light bulbs and fluorescent
lamps are incoherent sources.
The conditions for two sources of light to produce interference:
1. The sources must be coherent, i.e. there must be a constant phase difference between
them. The phase difference may be zero but it does not have to be.
2. The waves that are interfering must have approximately the same amplitude otherwise the
resulting interference pattern lacks contrast.
3. The two waves must have exactly same frequency and wavelength. With the waves of
different wavelengths, the position where the crests coincided would always be changing.
Young’s Double Slit Experiment:
In 1801 the English scientist Thomas Young (1773 – 1829) performed an experiment that
demonstrated the wave nature of light by showing that two overlapping light waves interfered
with each other. By this experiment he was also able to determine the wavelength of light.
Setup of the experiment:
In an experiment the light of single
wavelength (monochromatic light)
passes through a single narrow slit S0
and falls on two closely spaced, narrow
slits S1 and S2. These two slits act as
coherent sources of light waves that
interfere constructively and
destructively at different points on the
screen to produce patterns of alternative
bright (B) and dark (D) fringes. The
purpose of single slit is to ensure that
only light from one direction falls on the
double silt. Without it, light coming
from different points on the light source
would strike the source from different
directions and cause the patterns on the
screen to be washed out. The slits S1 and S2 act as coherent sources of light waves because the
light from each originates from the same primary source – the single slit.
A series of alternately bright and dark bands (interference fringes), which are equally spaced and
parallel to the slits, can be observed on a screen placed anywhere in the region of overlap.
Calculation of fringe separation
The opposite figure represents the relative positions of the
coherent sources, SI and S2, and a point P on the screen.
The perpendicular distance, D, from the plane of the slits
to the screen is very much greater than the slit separation, a
(typically, D = 20 cm, a = 0.1 cm). The distance travel by
waves along S2P is greater than the distance travel by
waves along SIP. The difference of two distances is called
path difference. By applying the mathematical calculation
using Pythagoras theorem we find that
−
=
Prepared by Faisal Jaffer, revised on 7-Mar-10
where x is the distance from the centre of the screen to the nth bright fringe. If a crest leaves SI at
the same time as a crest leaves S2, there will be a bright fringe wherever the path difference
(S2P – S1P) is equal to a whole number n of wavelength
−
=
By combining the above two equations
=
=
where n= 0, 1, 2, 3… for bright fringes.
Similarly for dark fringes the path difference is equal to the odd number of half wavelength
( + ).
=( + )
Typical values for the equation are D = 20cm = 0.2m
a = 1mm = 10-3m
λ = 6 x 10-7m
x = 0.12 mm=1.2 x 10-4m
By measuring the value of ‘D’ and ‘a’ and determining the value of ‘x’ by traveling microscope,
we can find the wavelength (λ) of the light.
The following points should be observed in double slit experiment.
1. The separation of the fringes can be increase by increasing ‘D’.
2. The separation of the fringes decrease by increasing ‘a’.
3. Increasing the width of any of the three slits increases the intensity of the fringes but more
blurred.
4. Moving single slit S closer to S1 and S2 increases the intensity of the fringes but does not
affect their separation.
Dark and bright fringes
To better understand the dark and light fringes consider the figure which shows three places on
the screen where the interference happened.
1. Figure (a) shows how the bright fringe arises directly opposite the midpoint between the
two slits. Here the light from the two slits travel exactly same distance and l1 is equal to l2 ,
therefore each containing the same number of wavelengths and constructive interference
happened.
2. In figure (b) the distance l2 is larger than l1 by exactly one wavelength and therefore a
bright fringe appear. Similarly whenever the difference between l1 and l2 is equal to
complete number of wavelengths that is λ, 2λ, 3λ … the constructive interference will
happen.
AS Physics Unit 5 - Waves
9
3. Figure (c) shows how the first dark fringe arises. Here the distance l2 is larger than l1 by
exactly one-half a wavelength, so the waves interfere destructively, giving rise to the dark
fringe. Wherever the difference between l1 and l2 equals to one half of wavelength the
destructive interference will happened.
THE DIFFRACTION GRATING
A diffraction grating is an arrangement of a large number
of parallel lines of equal width ruled on glass. In
diffracting grating there are clear spaces between the
rulings of about 600 lines per millimeter. Diffraction
gratings are used to produce optical spectra.
The figure represents a section of a diffraction grating
which is being illuminated normally by light of
wavelength λ.
Each of the gap acts like a very narrow slit and diffracts
the incident light to an appreciable extent in all the
forward directions. Consider that light which is diffracted
at some angle  to the normal. The slits are equally
spaced and therefore if  is such that light from all the gaps are in phase.
Thus, if the light from one slit interferes constructively with the light from the other slit
whose values of  would be given by the equation
=
Where n = 0, 1, 2, 3….and d is the distance between the two
lines in meters.
The effect of the grating, therefore, is to produce a series of
bright images on the screen, known as principal maxima, the
angular positions , of which are given by the above equation.
Each value of  applies to either side of the normal.
For any value of d and λ the zero order principal maxima, is
always at the center of the screen and given by replacing n = 0
The positions of the first order (n = 1), second order (n = 2), etc. principal maxima;
however, depend on both d and λ.
A typical grating for example has a grating spacing, d, of 1/600000 m (corresponding to 600
lines per millimetre, i.e. 600000 lines per metre). If such a grating is illuminated by light
whose wavelength,
λ is 6×10-7m, we find, by substituting these values in equation, that
n = 1 then  = 21.1o
n = 2 then  = 46.1o
but when n = 3 the equation gives sin  = 1.08 (which is impossible) therefore this particular
grating cannot produce a third order image when illuminated by light of this particular
wavelength.
It follows from equation that the number of orders of principal maxima that can be produced
can be increased by increasing d (i.e. reducing the number of lines per metre). Reducing the
total number of lines decreases the sharpness of the principal maxima and also gives rise to
faint images, subsidiary maxima, in the regions between the principal maxima.
Prepared by Faisal Jaffer, revised on 7-Mar-10
STANDING OR STATIONARY WAVES:
A stationary or standing wave results when:
1. two progressive waves which are traveling in
opposite direction,
2. which have the same speed and frequency and
approximately equal amplitudes,
3. are in same medium
4. and are overlapped or superimposed.
5. Standing waves can be produced by the
reflection of incoming waves on any hard
surface. In this process the phase of reflecting
wave becomes reverse of incident wave.
6. Places of maximum amplitude are called antinodes (A) and places of zero amplitude are
called nodes (N).
Standing waves in string:
Consider a string of length L
that is fixed at both ends. The
string has a set of natural
patterns of vibration called
normal modes. This can be determined very simply. First remember that the ends are fixed so they
must be nodes. This means a certain number of wavelengths or half wavelengths can fit on the
string determined by the length of the string. The first three possible standing waves are shown
below.
The wavelengths of the standing wave in string can be related by the length of the string. The
frequency, then, is found from the wavelength. v is the speed of wave in the string.
AS Physics Unit 5 - Waves
11
For nth harmonic frequency we the general term as:
=
=
2
where n= 1, 2, 3,....
The lowest frequency is called the 1st harmonic mode or the fundamental frequency. The higher
frequencies are called overtones. Multiples of the 1st harmonic modes are labeled as the 2nd, 3rd
and 4th harmonic modes.
Standing waves in air columns
Just as we have standing waves on strings we can have standing sound waves in columns of air.
The organ pipe is the basic example. At the open end of a pipe we expect displacement antinodes.
If the end is closed we would expect a displacement node. Using the same sort of arguments as
we did for the string we can find the normal modes of the air column in the pipe.
For two open ends the first four harmonics are:
=2×
=
Which is same as it is for two fixed end string.
2
Prepared by Faisal Jaffer, revised on 7-Mar-10
For air column closed at one end and open at the other the first four harmonics are:
Notice that the "closed at one end" case only exhibits the odd harmonics. The frequencies of the
normal modes for the two cases can be written as:
=
4
(2 − 1)
(2 − 1)
×
4
A good reference website can be looked for further study
http://openlearn.open.ac.uk/mod/resource/view.php?id=289475
=
Speed of sound wave in air column:
The speed with which sound travels in any medium may be determined if the frequency and the
wavelength are known. The relationship between these quantities is
=
where λ is the wavelength, f is the frequency and v is the speed of sound waves.
In this experiment the velocity of sound in air is to be found by using tuning forks of known
frequency. The wavelength of the sound will be determined by finding out the position of nodes
and antinodes produced in the air column above the water (resonance).
The apparatus for the experiment consists of a long cylindrical plastic tube attached to a water
reservoir. The length of the water column may be changed by raising or lowering the water level
while the tuning fork is held over the open end of the tube. Resonance is indicated by the sudden
increase in the intensity of the sound when the column is adjusted to the proper length. The
resonance is a standing wave phenomenon in the air column and occurs when the column length
is /4, 3/4, 5/4 where  is the sound wavelength.
AS Physics Unit 5 - Waves
13
The water surface constitutes a node of the standing wave since the air is not free to move
longitudinally. The open end provides the conditions for an antinode, but the actual antinode has
been found to occur outside the tube at a distance of about c=0.6r from the end, where r is the tube
radius. This end correction ‘c’ may be added to get a more accurate value if only one resonance
can be measured, but it is usually more convenient to eliminate this "end effect" by subtracting the
resonance length for /4 from those for 3  /4, 5 /4, etc.
4−−−−−−−(
+ = 3 4−−−−−−−(
Subtract equation 1 from equation 2
3
− =
−
4 4
+ =
−
1)
2)
=
2
= 2( − )
From the above equation the speed of sound waves can be found by substituting the equation in
=
=2 ( − )
Finding out the speed of sound waves using cathode ray
oscilloscope:
This requires practical demonstration in class.
Prepared by Faisal Jaffer, revised on 7-Mar-10
Exercise:
Q1. What is the phase difference between two waves, each of wavelength 12cm when one
leads the other by a) 6cm, b) 3 cm c) 9cm d) 12cm e) 14cm f) 18 cm g) 36 cm h) 39 cm.
Q2. The distance between the 1st bright fringe and 21st bright fringe in a Young’s double slit
arrangement was found to be 2.7mm. The slit separation was 1.0 mm and the distance
from the slit to the plane of the fringes was 25cm. What was the wavelength of the light?
Q3. In a Young’s double-slit experiment a total of 23 bright fringes occupying a distance of
3.0mm were visible in the traveling microscope. The microscope was focused on a plane
which was 31cm from the double slit and the wavelength of the light being used was 5.5 x
10-7 cm. What is the separation of the double slit?
Q4. When a grating with 300 lines per millimeter is illuminated normally with a parallel beam
of monochromatic light a second order principal maximum is observed at 18.9o to the
straight through direction. Find the wavelength of the light.
Q5. How many principal maxima are produced when a grating with spacing of 2.00 x 10-6 m
is illuminated normally with light of wavelength 6.44 x 10-7m?
Q6. Suppose that a string is 1.2 meters long and vibrates in the first, second and third
harmonic standing wave patterns. Determine the wavelength of the waves for each of the
three patterns.
Q7. The string at the right is 1.5 meters long and is vibrating
as the first harmonic. The string vibrates up and down
with 33 complete vibrational cycles in 10 seconds.
Determine the frequency, period, wavelength and speed
for this wave.
Q8. The string at the right is 6.0 meters long and is vibrating as the
harmonic. The string vibrates up and down with 45 complete
vibrational cycles in 10 seconds. Determine the frequency,
period, wavelength and speed for this wave.
Q9. The string at the right is 5.0 meters long and is vibrating as
the fourth harmonic. The string vibrates up and down with 48
complete vibrational cycles in 20 seconds. Determine the
frequency, period, wavelength and speed for this wave.
Q10.
The string at the right is 8.2 meters long and is
vibrating as the fifth harmonic. The string vibrates up and
down with 21 complete vibrational cycles in 5 seconds.
Determine the frequency, period, wavelength and speed for
this wave.
third