PROGRESSIVE WAVES

STATIONARY/STANDING WAVES
A stationary wave is a wave formed when two progressive waves of the same
type, amplitude and wavelength, travelling in opposite directions in the same
medium, superpose.

In a stationary wave, some points are permanently at rest whilst others are
vibrating with varying amplitude
Points which are permanently at rest are called nodes (N) and their amplitude is
zero.
Points which vibrate with maximum amplitude are called antinodes (A).
Antinodes are midway between the nodes
The distance between a node and its neighbouring antinode is one quarter of a
wavelength
Adjacent nodes (or adjacent antinodes) are separated by a distance equal to half
a wavelength.

Properties of Stationary Waves

 Energy is localized ie energy is not transferred away from source.
 Wave profile is stationary/standing
 Particles between successive nodes vibrate in phase, while particles in
neighbouring antinodes are in antiphase.
 Amplitude of particles varies from zero at nodes to maximum at
antinodes ie amplitude varies.
A string is attached at one end to a vibration generator, driven by a signal generator (Figure
15.6). The other end hangs over a pulley and weights maintain the tension in the string. When
the signal generator is switched on, the string vibrates with small amplitude. Larger
amplitude stationary waves can be produced by adjusting the frequency.

The pulley end of the string cannot vibrate; this is a node. Similarly, the end attached to the
vibrator can only move a small amount, and this is also a node. As the frequency is increased,
it is possible to observe one loop (one antinode), two loops, three loops and more.
A flashing stroboscope is useful to reveal the motion of the string at these frequencies
This experiment is known as Melde’s experiment, and it can be extended to investigate the
effect of changing the length of the string, the tension in the string and the thickness of the
string.

A string is stretched between two fixed points separated by a fixed distance L, for example a
guitar string. Pulling the middle of the string and then releasing causes the string to vibrate
and produce a stationary wave. Releasing the string produces two progressive waves
travelling in opposite directions. These are reflected at the fixed ends. The reflected waves
combine to produce the stationary wave. There is a node at each of the fixed ends and an
antinode in the middle.
Stationary waves of various frequencies can be set up in a string/wire by plucking the
string/wire at different points along the string/wire.

The string appears as a series of loops, separated by nodes. In the diagram below, point A is
moving downwards. At the same time, point B in the next loop is moving upwards. The
phase difference between points A and B is 180°. Hence the sections of spring in adjacent
loops are always moving in antiphase; they are half a cycle out of phase with one another
 At time t = 0, the progressive waves travelling to the left and right are in phase. The
waves combine constructively, giving an amplitude twice that of each wave.
 After a time equal to one-quarter of a period (t = T/4), each wave has travelled a
distance of one quarter of a wavelength to the left or right. Consequently, the two
waves are in antiphase (phase difference = 180°). The waves combine destructively,
giving zero displacement.
 After a time equal to one-half of a period (t = T/2), the two waves are back in phase
again. They once again combine constructively.
 After a time equal to three-quarters of a period (t = 3T/4), the waves are in antiphase
again. They combine destructively, with the resultant wave showing zero
displacement.
 After a time equal to one whole period (t = T), the waves combine constructively. The
profile of the spring is as it was at t = 0.
DIFFRACTION

WAVEFRONT
Properties of waves include reflection, refraction, diffraction and interference.
Diffraction is the spreading of a wave as it passes through an aperture/narrow opening or
around an edge.
The wave will try to curve around the boundary of obstacle or to curve outward through the
opening due to friction.
The size of the obstacle or opening must be comparable to wavelength of the wave
The extent of the diffraction effect is dependent on the relative sizes of the aperture to the
wavelength of the wave.
 More diffraction if the size of the gap is similar to the wavelength
 More diffraction if wavelength is increased (or frequency decreased)

The extent to which waves spread out depends on the relationship between their
wavelength and
the width of the gap. In a, the width of the gap is very much greater than the wavelength
and there is hardly any
noticeable diffraction. In b, the width of the gap is greater than the wavelength and there
is limited diffraction.
In c, the gap width is approximately equal to the wavelength and the diffraction effect is
greatest.
DIFFRACTION OF WATER WAVES
A ripple tank can be used to show diffraction. Plane waves are generated using a
vibrating bar, and move towards a gap in a barrier (Figure 14.6).

Where the ripples strike the barrier, they are reflected back. Where they arrive at
the gap, however, they pass through and spread out into the space beyond.
It is this spreading out of waves as they travel through a gap (or past the edge of
a barrier) that is called diffraction.
The extent to which ripples are diffracted depends on the width of the gap. This
is illustrated in diagrams a, b and c above.
The lines in this diagram show the wavefronts. It is as if we are looking down on
the ripples from above, and drawing lines to represent the tops of the ripples at
some instant in time.
The separation between adjacent wavefronts is equal to the wavelength λ of the
ripples.
When the waves encounter a gap in a barrier, the amount of diffraction depends
on the width of the gap. It is greatest when the width of the gap is roughly equal
to the wavelength of the ripples.

DIFFRACTION OF SOUND AND LIGHT WAVES

Diffraction effects are greatest when waves pass through a gap with a width
roughly equal to their wavelength. This is useful in explaining why we can
observe diffraction readily for some waves, but not for others.
For example, sound waves in the audible range have wavelengths from a few
millimetres to a few metres. Thus we might expect to observe diffraction effects
for sound in our environment.
Sounds, for example, diffract as they pass through doorways. The width of a
doorway is comparable to the wavelength of a sound and so a noise in one room
spreads out into the next room.
Visible light has much shorter wavelengths (about 5 × 10 -7 m). It is not diffracted
noticeably by doorways because the width of the gap is a million times larger
than the wavelength of light. As a result, a person must have a direct line of
sight to detect the light waves.
However, we can observe diffraction of light by passing it through a very narrow
slit or a small hole. When laser light is directed onto a slit whose width is
comparable to the wavelength of the incident light, it spreads out into the space
beyond to form a smear on the screen. An adjustable slit allows you to see the
effect of gradually narrowing the gap.

DIFFRACTION OF RADIO AND MICROWAVES

Radio waves can have wavelengths of the order of a kilometre. These waves are
easily diffracted by gaps in the hills and by the tall buildings around towns and
cities.
Microwaves, used by the mobile phone network, have wavelengths of about 10
cm. These waves are not easily diffracted (because their wavelengths are much
smaller than the dimensions of the gaps) and mostly travel through space in
straight lines.
Cars need external radio aerials because radio waves have wavelengths longer
than the size of the windows, so they cannot diffract into the car. If you try
listening to a radio in a train without an external aerial, you will find that FM
signals can be picked up weakly (their wavelength is about 3 m), but AM signals,
with longer wavelengths, cannot get in at all.
Example
A microwave oven (Figure 14.9) uses microwaves with a wavelength of 12.5 cm.
The front door of the oven is made of glass with a metal grid inside; the gaps in
the grid are a few millimetres across.
Explain how this design allows us to see the food inside the oven, while the
microwaves are not allowed to escape into the kitchen (where they might cook
us).

Solution
The grid spacing is much smaller than the wavelength of the microwaves, so the
waves do not pass through. However, the wavelength of light is much smaller, so
it can pass through unaffected.
Explaining diffraction
Diffraction is a wave effect that can be explained by the principle of
superposition. We have to think about what happens when a plane ripple
reaches a gap in a barrier.
Each point on the surface of the water in the gap is moving up and down. Each of
these moving points can be thought of as a source of new ripples spreading out
into the space beyond the barrier. Now we have a lot of new ripples, and we can
use the principle of superposition to find their resultant effect. Without trying to
calculate the effect of an infinite number of ripples, we can say that in some
directions the ripples add together while in other directions they cancel out
SINGLE-SLIT DIFFRACTION
Waves passing through a long narrow slit of width a produce, on a screen, a
diffraction pattern that consists of a central maximum (bright fringe) and other
lateral maxima
Maxima are separated by minima (dark fringes) that are located relative to the
central axis by angles θ:
Minima or dark spots are due to destructive interference of wavelets.
The maxima are located approximately halfway between the minima or the dark
spots.

Wavefronts from monochromatic light is made to be incident on a slit of finite

width a, as shown in the diagram below
Each point on the wavefront through the slit will acts as a source of light
wavelets. The wavelets at some points on the screen interfere destructively and
on other points they interfere constructively forming a diffraction pattern of
alternate bright and dark fringes.
To find the dark fringes, the slit is divided into upper and lower halves. The rays
in these halves are paired up into ray r1 from the top point of the upper half and
ray r2 from the top point of the lower half.
Wavelets along these two rays cancel each other when they arrive at P1.
Diagram below shows the pairing of the rays.
Path Length Difference:
The wavelets of the pair of rays r1 and r2 are in phase within the slit because
they originate from the same wavefront passing through the slit, along the width
of the slit. To produce a dark fringe, they must be out of phase by λ/2 when they
reach P1.
This phase difference is due to their path length difference (r2 – r1), with the
path travelled by the wavelet of r2 to reach P1 being longer than the path
travelled by the wavelet of r1.

This equation gives the values of θ for which the diffraction pattern has zero light
intensity— that is, when a dark fringe is formed as shown below
  If a narrow slit is used, say when a is halved, from the equation

1. the angle ɵ increases, and

2. the width of the central maximum is doubled

However, the intensity of the bright fringes decreases

Example1 A slit of width a is illuminated by white light. Calculate the value of
slit width a that will make the first minimum for red light of wavelength λ =
650nm to appear at θ = 15°.
Example 2 Light of wavelength 580 nm is incident on a slit having a width of
0.300 mm. The viewing screen is 2.00 m from the slit. Find the positions of the
first dark fringes and the width of the central bright fringe.
Example 3

Intensity/Brightness of Single-Slit Diffraction

The general features of the intensity distribution are shown in the diagram
above.
 The central fringe is the widest and brightest fringe while lateral/side
fringes are much weaker and much less bright fringes.
 Brightness/intensity of fringes decreases sideways
The central bright maximum is twice as wide as the secondary maxima/lateral
maxima

DIFFRACTION GRATING
A diffraction grating consists of many parallel lines ruled closely on a piece of glass or plastic
The ruled widths are opaque to light while the space between any two successive lines is transparent
and act as parallel slits
Each transparent line is capable of diffracting the incident light. When light is shone through this
grating, a pattern of interference fringes is seen.
A diffraction grating may have between 1000lines/cm to 5000lines/cm (100lines/mm – 500lines/mm)
The effect of the grating is to produce a series of bright images, or lines, at different angular position
The lines or maxima fringes are given order numbers from the central maxima n= 0 to the
lateral 2 sides of the central maxima.as shown below.
The Diffraction Grating with White light

 For the other orders of diffraction, each wavelength making up white light is diffracted by
amount, as described by the expression dsinθ = nλ. This is because the various components of
white light have different wavelength.
 Red light, because it has the longest wavelength, is diffracted through the largest angle and
blue with the shortest wavelength, is diffracted least.
 Thus white light is diffracted into its component colours, producing a spectrum as shown
below.
 The spectrum is repeated in different orders of the diffraction pattern.
Solution
From dsinθ = nλ,
sinθ = nλ/d = 1
n = d/λ ,
INTERFERRENCE
Interference is the superposition of waves from two coherent sources.
Coherent sources
Two sources are coherent when they emit waves that have a constant phase
difference. (This can only happen if the waves have the same frequency or
wavelength. So coherent sources have same frequency or wavelength
A laser is a coherent source but a light bulb is not.
Superposition of waves can cause constructive interference or destructive
interference
Constructive interference - Waves arrive at these points in phase resulting in
high amplitude hence high intensity, (regions of maxima)
Destructive interference - Waves arrive at these points in anti-phase resulting
in low or zero amplitude hence low or zero intensity, (regions of minima)

Interference of sound waves

A simple experiment shows what happens when two sets of sound waves meet.
  Two loudspeakers are connected to a single signal generator (Figure
14.11).
 They each produce sound waves of the same wavelength.
 Walk around in the space in front of the loudspeakers; you will hear the
resultant effect
 At some points, the sound is louder than for a single speaker.
 At other points, the sound is much quieter.
 The space around the two loudspeakers consists of a series of loud and
quiet regions.
 We are observing the phenomenon known as interference.
The loudspeakers are emitting waves that are in phase because both are
connected to the same signal generator. At each point in front of the
loudspeakers, waves are arriving from the two loudspeakers. At some points, the
two waves arrive in phase (in step) with one another and with equal amplitude.
We hear a louder sound.
At other points, something different happens. The two waves arrive completely
out of phase or in antiphase (phase difference is 180°) with one another. There is
a cancelling out, and the resultant wave has zero amplitude. At this point, we
would expect silence. At other points again, the waves are neither perfectly out
of step nor perfectly in step, and the resultant wave has amplitude less than that
at the loudest point.
Where two waves arrive at a point in phase with one another so that they add
up, we call this effect constructive interference. Where they cancel out, the
effect is known as destructive interference. Where two waves have different
amplitudes but are in phase, constructive interference results in a wave whose
amplitude is the sum of the two individual amplitude
.However, if the speakers were connected to different signal generators with
slightly different frequencies, the sound waves might start off in phase with one
another, but they would soon go out of phase (Figure 14.18).
We would hear loud, then soft, then loud again. The interference pattern would
keep shifting around the room.
Interference in a ripple tank
 The two dippers in the ripple tank (Figure 14.12) should be positioned so
that they are just touching the surface of the water.
 When the bar vibrates, each dipper acts as a source of circular ripples
spreading outwards.
 Where these sets of ripples overlap, we observe an interference pattern.
Another way to observe interference in a ripple tank is to use plane waves
passing through two gaps in a barrier. The water waves are diffracted at
the two gaps and then interfere beyond the gaps.

Figure 14.15 shows two sets of waves setting out from their sources.
  At a position such as A, ripples from the two sources arrive in phase with one
another, and constructive interference occurs.
 At B, the two sets of ripples arrive out of phase, and there is destructive
interference. Although waves are arriving at B, the surface of the water
remains approximately flat.

Whether the waves combine constructively or destructively at a point depends on the path
difference of the waves from the two sources.
 The path difference is defined as the extra distance travelled by one of the
waves compared with the other.
 At point A in Figure 14.15, the waves from the red source have travelled 3 whole
wavelengths. The waves from the yellow source have travelled 4 whole
wavelengths.
 The path difference between the two sets of waves is 1 wavelength.
 A path difference of 1 wavelength is equivalent to a phase difference of zero.
This means that they are in phase, so they interfere constructively.
 At point B, the waves from the red source have travelled 3 wavelengths; the
waves from the yellow source have travelled 2.5 wavelengths.
 The path difference between the two sets of waves is 0.5 wavelengths, which is
equivalent to a phase difference of 180°. The waves interfere destructively
because they are in antiphase

INTERFERENCE OF LIGHT
The Young double-slit experiment
 A source of monochromatic light is placed behind a narrow slit C.
 The single slit C sends out circular wavefronts
 Since the distance of the slits A and B from C are the same, the same wavefronts
arrive at A and B.
 The wavefronts from slit A and B are in phase and diffracts and spreads outwards
from each slit into the space beyond; That is, the double slits A and B act as coherent
sources of light.
 The spherical wavefronts overlap in the region of overlap after the double slits as
shown below

 A screen is placed in the region behind the double slits to capture the
overlapped/superposed spherical wavefronts.
 An interference pattern of light and dark bands called ‘fringes’ is formed on the
screen.
 If the superposed wavefronts are πrad/180o out of phase with each other, destructive
interference occurs, resulting in a dark patch on the screen and (region of minima) if
they are completely in phase, constructive interference occurs, resulting in a bright
patch on the screen (region of maxima).
 A pattern of bright and dark patches is observed on the screen. The bright and dark
patches are called interference fringes.