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6
How wavesbehave
96
Energy is conservedNow that we have a general idea of what a wave is and its basic
characteristics, we need to look at wave behaviour in more detail and
define different ways of representing wave behaviour using various
graphical methods.
We start with the fundamental principle of energy conservation.
This supports the entire discipline of physics (and was discussed
earlier in Section 4.4). The amount of energy in the universe is fixed,
so energy can be neither created nor destroyed. However, energy
may change from one form into another (energy transformation).
Waves are carriers of energy, and so they must be taken
into account when applying the principle of energy
conservation to systems in which waves are present.
6.1 Energy and wavesLet us consider a sound wave produced by a speaker in a science laboratory.
The speaker cone vibrates, pushing the air particles around it. The sound waves
propagate outwards in three dimensions from the speaker. They travel through
the air and eventually strike the walls, floor, windows and ceiling of the laboratory.
Let us look at this process from the point of view of energy. The energy used
to power the speaker is electrical energy, which is transformed into kinetic energy
as the speaker diaphragm wobbles back and forth. The kinetic energy is
transferred into the air particles in the room as the sound wave travels away from
the speaker. The energy spreads out into an increasing volume of space as the
wave propagates outwards from the speaker. Some energy is converted into heatin the speaker and the air. When the sound wave reaches aboundary, such as
the surface of a wall, some of the wave energy bounces back (is reflected), part of
it passes through (is transmitted) into the new medium and some of the energy is
lost as heat in the new medium (absorbed).
energy transformation, boundary,
intensity, inverse square law,
superposition, interference, phase,
constructive interference,
destructive interference, fixedboundary, free boundary, wave front,
ray, reflection, refraction, absorption,
law of reflection, incidence, normal
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loudmusic
Figure 6.1.1 Your parents can also enjoy the music you play in your bedroom. Some sound energyis reflected and some is absorbed; however, unfortunately for your parents, some
sound energy is transmitted through the walls and door.
If you stand next to the speaker, the sound is loud; as you move away, the
volume decreases. Outside the room, you can still hear the sound but it is much
softer and probably muffled (Figure 6.1.1). This is because the energy that
reaches your ears decreases as you move away from the source of the sound wave.
There are three main reasons for this decrease in energy with distance.
The first reason is that some of the original kinetic energy from the speaker
diaphragm is converted into other forms of energy by the media it travels
through. Some is dissipated (absorbed) as heat by the air molecules and the
materials that make up the floor, walls and ceiling. The second reason is that not
all of the sound wave makes it out of the room as some of it is reflected back
inside. The third reason is the inverse square law, which is discussed below.
So as a wave travels from its oscillating source, the energy carried by the wavedecreases; however, as the energy of the system must be conserved, we can account
for the apparently missing energy by considering the absorption and reflection of
energy at boundaries. Mathematically this can be represented as follows:
Ewave=Etransmitted+Ereflected+ Eabsorbed
The energy of a wave is proportional to the waves amplitude squared.
In sound waves, the amplitude is related to the volume (loudness) of the sound;
in light waves, it is related to the brightness of the light.
Ewave amplitude2
But even if the wave were to travel through a perfect medium, which doesnt
absorb and dissipate the wave energy as heat, the sound volume (or even lightbrightness) decreases as you move away from the source. The rate of energy
transfer by a source of waves through a given area is called the waves intensity.
Intensity is measured in watts per square metre (W m2). The rate of energy
transfer is called power, so wave intensity can be described using the following
equation:
Intensity=energy
time areaor Intensity=
power
areaor I=
P
A
Explain that the relationship
between the intensity of EM
radiation and distance from a
source is an example of the
inverse square law:
I1
2d.
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The energy density, and so the intensity, of a wave will decrease as you
move away from the source. Exactly how the intensity varies can be complicated
by many factors. The source, like a speaker, may mainly transmit the wave in one
direction and obstacles in the waves path may cause reflections and absorptions
of the wave energy. However, in the simplest case in which we assume that the
wave is transmitted uniformly in all directions with the mechanical energy
conserved as it spreads and we can ignore reflections and absorption, we can usethe inverse square lawto describe the variation of intensity with distance.
In this ideal case, all of the energy emitted by the source must pass through
the surface of a sphere with radius dmetres (Figure 6.1.2). The area of this
sphere will be 4d2, and the intensity of the wave at a point dmetres from the
source is given by the equation:
I=P
d4 2
d1 d
2
Figure 6.1.2 Energy produced by the speaker passes first through the surface of a sphere of radius d1,and then that same energy passes through the surface of the larger sphere of radius d2.
The equation below tells us that the intensity of a uniformly transmitted
wave with no mechanical energy loss decreases with the square of the distance d
from the source.
I12
d
In most cases, mechanical waves such as sound waves and water waves cannot
be accurately modelled using the inverse square law because energy is dissipated
as heat by the particles in the medium that the wave travels through. However,
electromagnetic (EM) waves do not require a medium to propagate and in airthere are practically no energy losses, so the inverse square law will predict
intensity levels for EM waves with high accuracy. For this reason, astrophysicists
use the inverse square law to compare and identify stars as there is little or no
energy loss in the vacuum of space.
Activity 6.1
PRACTICALEXPERIENCES
ActivityManual,Page48
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Worked example
QUESTION
The Sun produces EM waves that propagate through space to the Earth. The Sun has a
power output of 3.86 1026 W.
a Calculate the intensity of the Sun as seen from Earth. (d= 149 597 900 km)
b How does this compare with the intensity of the Sun seen from Jupiter,approximately 5 times the distance away?
SOLUTION
a Calculate the intensity, given that P= 3.86 1026 W and d= 149 597 900 km.
Convert all units into SI units: d= 149 597 900 1000 m.
I=3 86 10
26.
W
4 (149 5 97 900 1000) m2 2
=
P
4d2= 1372.5 W m2
The intensity at the Earth is 1370 W m2 (to 3 significant figures).
b Assume the distance from the Sun to the Earth is dmetres. Then the distance from
the Sun to Jupiter is 5dmetres. Therefore IEarth
1
2dandI
Jupiter
1
(5d)2 25d2
1,
sothe intensity at Jupiter will be1
25or 4% the intensity at Earth.
Figure 6.1.3 Sunset on Mars. Mars is3
2times further from the Sun than the Earth, so the setting
Sun appears2
3of the size on Earth and its intensity is
4
9that received on Earth.
CHECKPOINT 6.11 Outline five different energy transformations that can occur as light waves propagate from a source in a
science laboratory.
2 If the distance from a light source is tripled, what happens to the intensity of light as viewed from each point?
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6.2 SuperpositionThe concept of a wave was introduced in Chapter 5 as a vibration that transfers
energy from one place to another. The simplest mathematical representation of
waves are sine waves (y= sinx), and more complicated waves can be thought of
as combinations of different sine waves.This mathematical representation is very convenient and useful for physicists
in modelling and predicting wave behaviour. The ability to add different sine
waves together to model any complex wave situation arises because of a
fundamental property of wavessuperposition. Superposition is one important
property that distinguishes wave behaviour from particle behaviour.
Superposition is the amazing ability of two or more waves to combine
at the same point in space at the same time. Or to put it another way, the net
disturbance at any point in a medium is simply the sum of the separate waves
present. The superposition principle, which is a fundamental characteristic of
waves, was proposed by English physicist Thomas Young (17731829) in the
early nineteenth century (Figure 6.2.1).
This is simple to say and may notseem earth shattering, but consider what
would happen if we were to attempt
superposition with particles instead of
waves. Consider two tennis ballsit is
not physically possible for both tennis
balls to exist in exactly the same place at
exactly the same time (Figure 6.2.3). Try
it for yourself.
Now take two wavessay, crossed
beams of light from two torches (Figure
6.2.4). These waves can exist in exactly
the same place at exactly the same time
and when they do, they combine (or
superimpose) to make a more complex
wave. When the waves move past this
meeting point, they emerge as the
original uncombined light beams.
The powerful significance of this
property of waves may escape you as it is
difficult to conceptualise waves when we
are so accustomed to a particle world. Just imagine for a moment that the tennis balls
in our previous example could superimpose like waves, what would this look like?
The incoming tennis balls would meet and combine into a larger, probablyoddly shaped tennis ball. Then after the meeting place they would emerge as
single tennis balls again, indistinguishable from the original incoming balls
(Figure 6.2.5). There is also a more mathematical interpretation of the principle
of superposition, which is discussed in Section 6.4.
The term interference is used to describe the change in waves that occurs as
a result of superposition. The size and shape of the superimposed waves depend
on the amplitude, wavelength and frequency of the original waves. It also
depends on an additional wave propertyphase.
Describe the principle
of superposition.
THOMAS YOUNG
Thomas Young is considered to
be the father of physical optics
for his championing of the wave
theory of light and his explanation
of superposition. He was also a
talented linguist, learning Persian,
Arabic and Turkish. He used these
skills to translate some Egyptian
hieroglyphics using the Rosetta
Stone (Figure 6.2.2).
Figure6.2.1ThomasYoung
Figure 6.2.2TheRosettaStone
Figure 6.2.3 Two tennis balls unsuccessfullytry to occupy the same point in
space at the same time.
Figure 6.2.4 Light from two torches combinewhen they occupy the same
point in space at the same time.
Figure 6.2.5 Imaginary superposition oftwo tennis balls
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1
6.3 PhasePhase is the key to understanding how waves superimpose and interact with
media and boundaries. Waves displace the particles of the media they travel
through. Let us consider one particle in the medium. Sometimes the particle is
displaced a maximum positive amount (crest) from its original position, sometimes
it is displaced a maximum negative amount (trough) and sometimes it is in its
original position (equilibrium). This means a particle is displaced by the wave in
a regular cycle: crest equilibrium trough equilibrium crest and so
on. The phase of a wave can be thought of as a label for the part of the
cycle that the particle is undergoing at a given time.Since we are using a sine function such asy= sin (x) to represent our wave,
the simplest way to label which part of the cycle the oscillating particle is in is to
state the value in brackets (x) (mathematically speaking, the argument). Since
the sine function comes traditionally from trigonometry, this value (the phase) is
normally given in angle units, such as radians or degrees; however, the phase is
not really an angle, just a mathematical label (Figure 6.3.1).
The idea of phase is easier to grasp when we think of the phase of two waves
relative to each other. If two waves cause a particle to be displaced the same
direction at the same time, they have a phase difference of 0 and are said to be
in phase. If the phase difference is 180 or radians, the waves are said to be
exactly out of phase (Figure 6.3.2).
Waves in phase Waves exactly 180
out of phase
Waves out of phase by
approximately 90
Figure 6.3.2 Waves in and out of phase
CHECKPOINT 6.31 Draw a diagram of two waves that have equal amplitude and frequency but are out of phase by 270 or
3
2
radians.
PHASE AND THEWAVE EQUATION
We have been using a very
simple equation, y= sin (x),
to describe wave behaviour. A more
powerful and useful description
requires a function that relates
horizontal displacement (x),
vertical displacement (y) and time
(t), and contains all the important
properties of that wave:
y A x ft= sin( )2
2
where A is the wave amplitude,
is the wavelength and fis
the frequency.
When using this equation to
describe a wave, the phase ()
of an individual wave is the
argument of the sine function:
= 2 2
x ft
CHECKPOINT 6.21 Define the concept of superposition.
2 Identify two properties common to both particles and waves.
Amplitudeoftheparticle
0 45 90 135 180 225 270 315 360
0 2 P32
2P
Deg
Rad
Phase (in degrees and radians)
P
Figure 6.3.1 The particle at point Phas a
phase of 45 or
4radians.
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6.4 The superposition of wavesHere is a more mathematical interpretation of the superposition principle. It says
that when two waves cross the same part of space at the same time, the resulting
wave is simply the mathematical sum of the two original waves.
We can use a graphical method for superimposing two waves in sine form.We plot the waves on the same axes, accurately recording the amplitude, frequency
and phase. Then moving from right to left, at every value ofxwe simply add the
corresponding heightstheyvaluesof the two sine waves. When adding the
heights, remember that theyvalues above the axis are positive and those below
the axis are negative (Figure 6.4.1).
1.5
1.0
0.5
0
0.5
1.0
1.5
0 90 180 270 360
Phase ()
Amplitude(m) w1
w2
ws
Figure 6.4.1 Two sinusoidal waves (w1 and w2) with different amplitudes and frequencies travelfrom left to right. The waves superimpose to give the resultant wave w
s.
This procedure can be carried out for any two waves. However, two special
cases emerge when superimposing waves of the same frequency and amplitude
(Figure 6.4.2). If we superimpose two such waves that are in phase, we see
a resulting maximum disturbance in the medium; to be exact, the resulting wave
will have double the amplitude of either of the original waves. This is called
constructiveinterference. If we superimpose two waves that are exactly
180 out of phase, we see a resulting zero disturbance in the medium. The waves
cancel each other out completely, the resulting amplitude is zero and so no
oscillation of the medium is observed. This is called destructive interference.
w1
w2
ws
w1
w2
ws
Constructive interference
Destructive interference
Figure 6.4.2 Two identical sinusoidal waves(w1 and w2) travel from left to
right. They superimpose to give
the resultant wave ws.
Constructive interference
occurs when the phase
difference is 0 (0 radians),
and destructive interference
occurs at 180 ( radians).
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Adding two waves together using a graphical method is relatively
straightforward, but adding three or more waves together in this way becomes
extremely time-consuming. A mathematical technique called Fourier analysis and
synthesis allows multiple waves to be added quickly and easily. For example,
electronic music and voice recognition software use Fourier analysis and synthesis
to add and subtract sound waves to create and recognise a wide variety of sounds
(See Physics Feature Beautiful mathematics and electronic music on page 104).When waves reflect from a boundary between two media, the phase of the
reflected wave depends on the nature of that boundary. There are two types:
fixed boundaries or free boundaries (Figure 6.4.3).
A fixed boundary has particles that are unable to oscillate, an example of
which would be a rope tied securely to a wall. If you wiggle the free end of the
rope, a transverse wave will travel down the rope towards the fixed boundary at
the wall. The wave will then be reflected from that boundary. The reflected wave
will be exactly out of phase with the original wave. This is because the rope is
tied at the wall and must always have a displacementy= 0 at that point. While
they overlap, the original wave and its reflection can be thought of as two
interfering waves. Any overlapping waves must superimpose to give zero
displacement at the wall. This can only occur when the original and reflectedwaves are exactly out of phase (phase difference of 180).
In a free boundary the particles in the adjacent media are free to move, so
waves transmitted through or reflected from free boundaries have the same phase
as the original wave.
Reflection from a fixed boundary Reflection from a free boundary
a b
Figure 6.4.3 (a) Waves are reflected from a fixed boundary exactly out of phase; (b) a freeboundary reflects the wave in phase.
Activity 6.2
PRACTICALEXPERIENCES
ActivityManual,Page53
DESTRUCTIVECAN BE USEFUL
In some factories where loud,
repetitive noise is a problem,
workers can wear special
headphones that sample the
surrounding noise and then
replay into the workers ears a
copy of this noise with exactly
the same amplitude but exactly
180 out of phase with it. The
result is destructive interference,
which means no noise reaches
the workers ears. This is called
anti-phase noise reduction.
However, since this effectdoesnt work very well with non-
repetitive noise such as human
speech, the workers are still able
to hear co-workers talking.
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104
PHYSICS FEATURE
BEAUTIFUL MATHEMATICS AND
ELECTRONIC MUSIC
The French mathematician Jean Baptiste Joseph,
Baron de Fourier (17681830) devised a
beautiful mathematical technique for synthesising a
waveform of any shape imaginable. His theory states
that any wave with a spatial frequency of fcan be
synthesisedby a sum of harmonic waves with
frequencies f, 2f, 3f, 4fand so forth. Any wave can
be thought of as a result of the addition of
overlapping sine and cosine waves.
Consider the example shown in Figure 6.4.4. The
waveform in Figure 6.4.4a is the result of combining
the six sine waves in Figure 6.4.4b. These six sinewaves with different frequencies are called the
harmonics. The frequency of the resultant wave has
the same frequency as the first harmonic (f1). The
harmonics can be illustrated using a spectrum graph
like Figure 6.4.3c. This plots the amplitude of the
harmonic versus the frequency.
Electronically synthesised music utilises the
mathematics of Fourier. An audio engineer
programming an electronic synthesiser keyboard, for
example, would use a signal generator to produce the
harmonic sine waves. By manipulating the amplitudes,frequencies and phases of these sine waves, the
desired sound can be selected. Similarly, a natural
sound can be copied and electronically reproduced.
The waveform of the natural sound is analysed to
determine its harmonics, which can then be easily
reproduced using a signal generator and synthesised
when required.
a
b
f1
f2
f3
f4
f5f6
Harmonics
f1 f2 f3 f4 f5 f6
Frequency
Amplitude
c
Figure 6.4.4 (a) The synthesised waveform; (b) the six componentharmonics of (a); (c) a spectrum graph of the harmonics
CHECKPOINT 6.41 What phase difference is required for two waves to destructively interfere?
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6.5 Diagrams used to describe wavesIn addition to the equations and graphs we have been using to describe wave
behaviour, it is common to use two additional diagramswave fronts and
raysto illustrate wave behaviour in media and at boundaries between media.
Waves originate from an oscillating source. We imagine for simplicity that thesource is tiny, called a point source. In Figure 6.5.1, transverse waves move out in
two dimensions from the oscillation caused by a tiny vibrating source. If we draw
a line joining the peak of each of these transverse waves, we have constructed a
wave front. A wave front is therefore an imaginary line that joins points of
equal phase. The concentric circular lines (ripples) that you see on the disturbed
surface of a pond are wave fronts.
For waves that propagate in three dimensions, the wave front would be a
spherical surface joining points of equal phase. The distance between two
adjacent wave fronts is one wavelength. Wave fronts that are closer to a source
appear more curved. As the wave travels a large distance from the source, the
wave fronts appear as parallel lines (called plane waves). A wave of a fixed
frequency travelling through a uniform medium will have wave fronts of equalspacing. The greater the frequency, the closer the spacing of the wave fronts.
Superposition is illustrated by overlapping wave fronts (Figure 6.5.2). Where
the wave fronts overlap, we have two waves combining with the same phase. At
this point there would be constructive interference.
Figure 6.5.2 Overlapping ripples from two disturbances on a water surface. The ripples are wavefronts, and superposition of the two waves occurs where two wave fronts overlap.
An imaginary line drawn perpendicular to a wave front in the direction
of propagation is called a ray (Figure 6.5.3). The ray is simply a line that points
in the direction that the wave front is moving. Rays are commonly used to show
the path of light through an optical system. Unlike wave fronts, rays do not giveany information about the wavelength or frequency of the wave.
CHECKPOINT 6.51 Define the terms wave frontand ray.
2 How does a wave front diagram give information about the wave frequency or wavelength?
Activity 6.3
PRACTICALEXPERIENCES
ActivityManual,Page59
Waves are emitted in all directions
from the light source.
An imaginary line drawn that joins points
of equal phase is called a wave front.
Figure 6.5.1 Constructing wave fronts fortransverse waves
Figure 6.5.3 A ray is drawn perpendicularthe wave front and shows the
direction of wave propagatio
wave fronts
ray
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6.6 Wave reflection and refractionAt the beginning of the chapter, we discussed the energy of a wave and what
happens at the interface between two media (a boundary). When a wave
encounters a boundary three things happen (Figure 6.6.1):
1 Part of the wave energy bounces off the interface and travels back into theoriginal mediaknown as reflection.
2 Part of the wave energy continues into the new mediaknown as
transmission or refraction.
3 Part of the wave energy is transferred to particles in the media as heat
known as absorption.
incident
refracted
reflected
Figure 6.6.1 Parallel light wave fronts incident on a surface (such as a piece of glass). Some ofthe light is reflected from the surface and some is refracted.
REFLECTINGHISTORY
The law of reflection was first
described by the Greek
mathematician Euclid in the
book Catoptrics, dated
approximately 200 BC. Catoptrics
is an ancient Greek term that
means reflection. The firstwritten description of a reflective
surface, a womans looking glass,
appears in Exodus 38 : 8, dated
approximately 1200 BC.
Figure 6.6.2 Anearlydepictionofa reflective
surfaceinart.Thisstone relief
is from the sarcophagusof
QueenKawitandshowsher
holdinga mirror, dated
approximately2061BC.
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1
ReflectionThe behaviour of reflected waves is described by the law of reflection.
This law states that the angle ofincidence equals the angle of reflection.
The angle of incidence (i) is the angle made by the incoming (incident) wave
front and the boundary. The angle of reflection (r) is the angle made by the
outgoing (reflected) wave front and the boundary (Figure 6.6.3). Therefore:
i=r
If a wave is normally incident on a boundary, then i=r= 0 and the wave
reflects back on itself.
A B
A B
A B
A B
mirror surface
mirror surface
mirror surface
mirror surface
Incident wave front
just reaching mirror
Reflected wave front
just leaving mirror
i
i
r
ir
r
a c
b d
Figure 6.6.3 The incoming (incident) wave front makes an angle of i with the reflective surface.The reflected wave front makes an angle of r with the mirror. The law of reflection
says i=r.
Wave front diagrams can quickly become cluttered, so it is usual to represent
the same concept concisely using rays (Figure 6.6.4). A large number of wave
fronts are replaced by an incident and reflected ray. The angles of incidence and
reflection are measured relative to the normal, which is a line drawn
perpendicular to the boundary.
incident ray reflected ray
normal
N
i r
Figure 6.6.4 Reflection of a wave using a ray diagram. The incident and reflected rays make anangle of i and r respectively, relative to the normal (N ).
Describe and apply the law of
reflection and explain the
effect of reflection from a pla
surface.
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108
RefractionImagine that a surf lifesaver is running up the hard sand near the water and is
then continuing on into the soft sand. As the medium changes from hard sand
to soft sand, the surf lifesaver slows down as it is harder to run in soft sand.
In the same way the speed of a wave changes as it moves from one
medium into another. If the wave encounters the boundary at an angle
(i 0), the wave fronts bend as they cross the boundary. This bending ofwaves across boundaries is called refraction. (See Figures 6.6.5 and 6.6.6.)
medium 1
a b
medium 2 medium 1 medium 2
i
i
r
r
vi
vi
vr
vr
i = 0
i
Figure 6.6.6 The wave slows down as it enters the second medium and so the wave fronts becomemore closely spaced. (a) The wave front is normally incident on the boundary (i= 0).
(b) The wave front encounters the boundary at an angle (i 0).
The bending is also evident when the waves path is represented by rays, as
shown in Figure 6.6.7. The incident ray travelling through medium 1 makes
an angle i (angle of incidence) with the normal, and the refracted ray through
medium 2 makes an angle ofr (angle of refraction) with the normal. If the
wave slows down on entering the new medium, the ray bends towards the normal
(i
>r
). If the wave speeds up the opposite occurs: the ray bends away from the
normal (i
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1
The degree to which a wave is refracted depends on the properties of the
media. The physical state, density, crystal structure and temperature of a
substance will affect the speed of the wave through that substance. The speed
of light waves is changed by the refractive index (n) of a substance, while the
acoustic impedance (Z) of a substance changes the speed of sound waves.
TRY THIS!MARCHING TO ILLUSTRATE
REFLECTION AND
REFRACTION
Link arms with some friends to
form a wave front. March in time
at the same speed. Reflect
yourselves from a flat surface,
such as a wall. As each person
reaches the wall, march backwards
at the same speed. Try this first
with the wave front parallel to the
wall and then at an angle. Then
reflect yourself from a curved
surface, like a curved gutter or
garden bed edge. You will see the
wave front shape change. To refract,
the marching speed needs to change as you change medium.Try marching from concrete onto grass. As the medium
changes, halve your speed. The wave front will bend if you
approach the boundary at an angle.
wall grass
concrete
Figure 6.6.8 (a) Students are reflected from the wall by marching backwards. (b) Studentsare refracted across the boundary by changing marching speed.
CHECKPOINT 6.61 Describe the law of reflection.
2 Define the concept of refraction.
a b
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6 PRACTICALEXPERIENCES
110
CHAPTER 6This is a starting point to get you thinking about the mandatory practical
experiences outlined in the syllabus. For detailed instructions and advice, use
in2 Physics @ Preliminary Activity Manual.
ACTIVITY 6.1: MODELLING THE INVERSE SQUARE LAWUse a light probe attached to a data logger or hand-held meter to measure light
intensity at different distances from a source.
Equipment list: a bright light source (lamp), light-sensitive probe or meter,
data logger, computer, tape measure.
metre ruler
photocell
light
source
light
sensorto computer
Figure 6.7.1 Experimental set-up for measuring light intensity at different distances
Discussion questions
1 Describe the relationship between light intensity and distance using the
data collected in this investigation. How does it compare with the inverse
square law?
2 Identifya possible source of experimental error in this investigation. What
strategies could you use to reduce the impact of the experimental error?
ACTIVITY 6.2: SUPERPOSITION OF WAVESUse a cathode ray oscilloscope (CRO) or computer program to observe the
superposition of pulses and waves.
Equipment list: cathode ray oscilloscope, 2 signal generators, graph paper,
computer.
256 Hz
256 Hz
signal
generator
signal
generator
cathode ray oscilloscope (or computer)
Figure 6.7.2 An oscilloscope connected to two signal generators
Plan, choose equipment or
resources for and perform a
first-hand investigation, and
gather information to model the
inverse square law for light
intensity and distance from a
source.
Perform a first-hand
investigation, gather, process
and present information using
a CRO or computer to
demonstrate the principle of
superposition for two waves
travelling in the same medium.
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Discussion questions
1 Explain the importance of phase difference to the superposition of two
waves with the same frequency and amplitude.
2 Describe the characteristics of the resultant wave when two waves of
different frequencies are superimposed.
ACTIVITY 6.3: WAVE FRONTS AND RAYSUse a light box and a variety of reflective surfaces to observe the reflection of light.
Draw accurate ray and wave front diagrams to show light reflection from plane,
concave and convex surfaces.
Equipment list: transformer, light box, plane mirror, concave mirror, convex
mirror, pencil, ruler, blank paper, protractor.
light box
plane mirror
convave and convex mirrors
Figure 6.7.3 A light box and reflective surfaces
Discussion questions
1 Explain how the shape of the reflective surface changes the shape of the
reflected wave front. Refer specifically to the law of reflection.
2 Describe the parts and function of the light box and explain how it
approximates a source a large distance away.
Perform first-hand
investigations and gather
information to observe the pa
of light rays and construct
diagrams indicating both the
direction of travel of the light
rays and a wave front.
Present information using ray
diagrams to show the path of
waves reflected from:
plane surfaces
concave surfaces
convex surfaces.
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Chaptersummary Energy is conserved in all systems.
Ewave=Etransmitted+Ereflected+Eabsorbed The energy of a wave is proportional to the amplitude
squared.
Intensity is defined as the rate of energy transfer
through a given area. It is measured in watts per square
metre (W m2). The intensity of a wave decreases with the square of the
distance from the source I12
d.
Superposition is a property that distinguishes waves
from particles.
The net disturbance at any point in the medium is the
sum of separate waves present.
The superimposed (or net) wave depends on the
amplitude, wavelength, frequency and phase of the
original waves.
Phase is the point in the cycle that an oscillating
particle is up to at a given time. Phase is a dimensionless quantity given as an angle in
degrees or radians.
Waves are in phase if the phase difference is 0 or
0 radians.
Waves are out of phase if the phase difference is
180 or radians.
The superposition of two waves in phase results in
constructive interference.
The superposition of two waves out of phase results in
destructive interference.
A wave front is an imaginary line joining points ofequal phase.
Wave fronts close to a source appear curved; at large
distances, they are parallel (called plane waves).
The distance between two adjacent wave fronts is
one wavelength.
A ray is an imaginary line drawn perpendicular
(at 90) to a wave front. The ray points in the direction
of propagation.
The law of reflection states that the angle of incidence
equals the angle of reflection (i=r).
A wave changes speed as it moves from one medium to
another. This is called refraction. Refraction causes wave fronts and rays to bend as they
cross the boundary from one medium to another.
The degree to which a wave is refracted depends on the
properties of the media.
PHYSICALLY SPEAKINGCreate a visual summary for the concepts in this chapter using a mind map.
1 Copy the table containing words, diagrams and equations.
2 Cut along the dotted lines so that you have 21 separate boxes.
3 Group related boxes together.
4 Stick the groups of boxes onto a sheet of blank paper.
5 Connect boxes with labelled links to form a mind map.
Amplitude Phase Wavelength
Constructive interference Ray I1
2d
Destructive interference Reflection i=r
Distance Refraction
Energy Superposition
Frequency Wave
Intensity Wave front
Reviewquestions
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REVIEWING 1 An aquarium has a light on top of the tank, as shown in Figure 6.7.4.
Draw and label the diagram to illustrate what happens to the energy of
the light waves as they propagate into the tank.
light
air pump
fish tank
Figure 6.7.4 An aquarium
2 The amplitude of a wave is doubled. Are the following statements trueor false?
a The wave frequency also doubles.
b The wave period also halves.
c The wave energy also quadruples.
d The wave speed also doubles.
3 Complete the table to show the relationship between intensity and distance.
DISTANCE d 2d 3d 4d 5d1
4d
1
2d
1
2d
INTENSITY I
4 Complete the table to show the relationship between degrees and radians.
DEGREES 30 90 180 270
RADIANS 0
4
32
5 Draw the rays corresponding to the wave front diagrams.
a b
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6 Draw the wave fronts corresponding to the ray diagrams.a b
7 Draw the wave fronts and rays as the wave is reflected from the boundary.
incident ray
mirror35
8 Draw the wave fronts and rays as the wave slows down on entering thenew medium.
medium 1 medium 2
normal
9 Samuel draws a ray diagram of a light beam reflecting from a planesurface. Use Samuels diagram to determine the angle of incidence and
the angle of reflection.
normal
mirror65
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SOLVING PROBLEMS10 A pulse is produced in a string of initial amplitude 35 cm. After the pulse
has travelled 1 m, its amplitude is 7 cm.
a Calculate the percentage of the original energy carried by the pulse
1 m from the source.
b Calculate the percentage of the original energy that has been lost.Can you account for the missing energy?
11 Helen purchases two light bulbs with power ratings of 40 W and 80 W.How far must she stand from the 80 W bulb so that it appears to have
the same intensity as the 40 W bulb?
12 Star A is twice as far away as Star B, but they generate the same light
intensity. Which star appears brighter and by what factor?
13 Stars C and D are both at a distance of 15 parsecs from Earth, butstar C is nine times brighter than D in the night sky. At what distance
would star C have to be in order to appear to be the same brightness as D?
14 A scuba divers underwater microphone detects a whale call 50 m awaywith an intensity of 0.47 mW m2. Another scuba diver is 1 km away at
another dive site. What will be the intensity at that distance? Ignoreabsorption losses.
15 Use graph paper to accurately reproduce these waves. Use the graphicalmethod to superimpose the waves and find the net disturbance.
a b
c d
Present graphical information
solve problems and analyse
information involving
superposition of waves.1.5
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