1
Combination wave
 In a material medium, the resorting force is provided by
intermolecular forces. If a molecule is disturbed, the
restoring forces exerted by its neighbors tend to return the
molecule to its original position, and it begins to oscillate.
In so doing, it affects adjacent molecules, which are in turn
set into oscillation. This is propagation of wave.
 Medium – the substance or object in which the wave is
travelling.
 A pulse: a single disturbance that travels through a
medium
 Why are waves important?  waves carry energy 
water wave
There are two types of waves regarding medium.
1. Mechanical waves
 Ones that need medium to propagate, where particles of
the medium oscillate as the wave passes through.
A mechanical wave is a disturbance that propagates
through a medium – solids, liquids or gases, thus
transferring energy from one place to another.
Travelling/Continuous/Progressive wave:
 continuous disturbance
 transfer energy from one place to another.
without a net motion of the medium through which they
travel.
 they all involve oscillations – SHM, of one sort or another.
 The important thing is that when a wave travels in a
medium, parts of the medium do not end up at different
places.
 The energy of the source of the wave is carried
to different parts of the medium by the wave.
Transverse wave
The particles of the medium oscillate perpendicular to
the direction of energy transfer/propagation of the wave.
 Earthquake secondary
waves, waves on a
stringed musical
instrument, waves on the
rope,
waves on strings
waves in water
● ocean waves
● sound waves – pressure waves in gas, solid or liquid
● in short, every wave that is NOT EM wave
● As the disturbance moves, the parts of the material
(segment of string, air molecules) execute harmonic
motion around equilibrium position
● Disturbance travels not the medium
●
●
2. Electromagnetic waves
The other ones, ELECTROMAGNETIC WAVES, do not need
medium to propagate. They come to us from faraway stars
traveling through a vacuum. Of course, they can travel
through a medium, but when they travel through medium,
they do definitely not make particles of the medium vibrate at
EM frequency. Just imagine window oscillating at frequency
of visible light, ~ 1015 Hz. On the other hand when a sound
wave (mechanical wave) travels through a window it will
make glass vibrate at that frequency.
 EM waves: light, radio waves, microwaves…
A wave of energy.
The electric and
magnetic field oscillate
(change magnitude
and direction)
Longitudinal/ Compression/ Pressure wave
The particles of the medium oscillate parallel to the
direction of energy transfer/propagation of the wave.
Sound waves in any medium, shock waves in an earthquake,
compression wave along a spring…
 Sound waves in
any medium, shock
waves in an
earthquake,
compression wave along the string
rarefaction – region in a medium with low pressure, low
density.
compression – region in a medium with high pressure,
high density.
More later
Speed of mechanical waves
The speed of the wave is the speed of energy transfer
and is not the same as the speed of the particle of the
medium oscillating around equilibrium position.
 The wave speed is determined by:
● the stiffness of the material
 more stiff  higher speed
each segment of medium is in tighter contact with its
neighbor
● density - more difficult to change the velocity of larger
masses than smaller ones
 greater density more inertia  lower speed
2
 How fast is transverse wave in strings?
Displacement vs. position graph shows the
displacement of all points along the wave.
A snapshot of a wave at one instant of time.
● T is tension in the string
 more tension  higher speed
● m/L is the mass per unit length of the string
 thicker rope  lower speed
v
Transverse wave:
displacement vs. x,
Longitudinal wave:
density vs. x or
pressure vs. x
T
m/ L
 Why do waves travel faster in steel than in air?
As far as waves are concerned, the difference between steel
and air is that steel is stiffer and denser than air.
But the stiffness of steel is much greater than that of air, even
though the density of steel is greater.
Consequently, the stiffness factor influences the wave speed
more and waves travel much faster in steel than in air.
Speed of sound in:
Displacement vs. time graph shows the oscillations
air: 343 m/s
helium: 1005 m/s
water: 1500 m/s
bone: 3000 m/s
glass: 4500 m/s
steel rod: 5000 m/s
Waves in a violin string: A-string: 288 m/s, G-string: 128 m/s
of one point on the wave. All other points will oscillate in
a similar manner, but they will not start their oscillations
at exactly the same time.
 Transverse waves cannot propagate in a gas or a liquid
because there is no mechanism for driving motion
perpendicular to the propagation of the wave.
Transverse wave:
displacement vs. t,
Longitudinal wave:
density vs. t or
pressure vs. t
Definitions associated with waves
Amplitude, A
● is the maximum displacement of a particle from its
equilibrium position.
● It is also equal to the maximum displacement of the
source that produces the wave.
● energy of a wave ∞ A2.
Period, T
Wave Equation
● is the time taken for one complete wave to pass any
given point.
distance
wavelength
=
time
period
= wavelength  frequency
wave speed =
Frequency, f
● the number of wavelengths passing by a given point.
f=
1
T
T=
1
f
Wavelength, λ
● This is the distance along the medium between two
successive particles that have the same displacement
(that are in phase – e.g. from crest to crest, or from
compression to compression)
Wave speed, v
● The speed at which wavefronts pass a stationary
observer.
● It is constant, depending on the medium only.
Energy, Power (Energy per time) and Intensity (Power
per unit area received by observer)
● Hence for a wave of amplitude A, we have that
Energy ∞ A2 , Power ∞ A2, Intensity ∞ A2
v=
λ
= λf
T
This applies to all waves  water waves, waves on strings,
sound waves, radio, light . .
Waves with different frequencies and wavelength will have
the same speed in one medium, determined by that medium.
If you shake the string faster (greater freq.) the wavelength
will be smaller and vice versa
3
Wave fronts propagating from a point source
● EM wave is made up of changing electric and magnetic
fields.
● The electric and magnetic field components of EM wave
are perpendicular to each other and also perpendicular to
the direction of wave propagation – hence EM waves are
transverse waves.
 Spherical wave – The center of the circle is the source
of the oscillations. If there is 3-D medium the wave will
spread out in all directions. And if the medium is uniform
these waves are spherical.
 Ray shows direction of wave/ energy propagation
 Wavefront is the set of crests at the same distance from
the source.
 Plane waves: far away from the source circular
wavefronts can be approximated with straight parallel lines
/ planes in 3-D. These are known as plane waves.
 EM waves striking the earth are plane waves
Electromagnetic waves –
Electromagnetic spectrum
● Visible light is one part of a much larger spectrum of similar
waves that are all electromagnetic.
● EM waves are produced/generated by accelerated
charges.
● They all travel travel through vacuum with the same speed
– speed of light c:
c = 2.99 792 458 x 108 m / s
c ≈ 3 x 108 m/s
● This speed is completely independent of the frequency or
the wavelength of the wave!!
● EM waves are waves, so: c = λf
greater λ smaller f
● The energy of a wave is directly proportional to its
frequency, but inversely proportional to its wavelength.
In other words, the greater the energy, the larger the
frequency and the shorter (smaller) the wavelength.
Short wavelengths are more energetic than long
wavelengths.
● Although all EM waves are identical in their nature,
they have very different properties, due to different
wavelengths and frequencies, and therefore energy
that they carry along.
4
Sound
 Sound is mechanical, longitudinal wave. Can be spread in
gases, liquids and solids.The wave consists of
compressed regions alternating with rarefied regions.
 The maximum (minimum) pressure during normal
conversation is 3  10 5 % higher (lower) than normal
pressure. Ear can detect such small changes.
 Just like a speed of a wave on a string, the speed of
sound is determined by the properties of the medium
through which it propagates. In air, under normal
atmospheric pressure and temperature, the speed of
sound is approximately 343 m/s.
 Frequency of the sound determines the pitch of the
sound. The pitch is perceived frequency of the sound.
 Humans’ audible range is 20 Hz – 20 kHz
 Infrasonic sound – frequencies below 20 Hz
 Ultrarasonic sound – frequencies above 20 kHz
 Dogs can detect frequencies as low as 50 Hz and as high
as 45,000 Hz while cats detect frequencies between 45 Hz
and 85,000 Hz. Bats who rely on reflection of sounds that
they emit for navigation can detect frequencies as high as
120,000 Hz. Dolphins can detect frequencies as high as
200,000 Hz. Infrasound in a range of 5 Hz to 10,000 Hz
can be detected by elephants.
 Infrasound is used in the nature for communication:
elephants (~ 15Hz) couple of kilometers, whales – as
sound travels faster in water (v ~ 1500 m/s) than in air, the
call can be heard over distances of thousands kilometers.
 Sources of infrasonic waves include earthquakes, thunder,
volcanoes, and waves produced by vibrating heavy
machinery. This last source can be particularly
troublesome to workers, for infrasonic waves – even
though inaudible – can cause damage to the human body.
These low freq waves act in a resonant fashion, causing
considerable motion and irritation of internal organs of the
body.
 Ultrasound is used for echolocation: dolphins, bats, sonar,
sonograms ....
 Sound needs a medium – won’t travel in a vacuum since
nothing to compress and expand
 Frequency is determined by the source of oscilations,
so when guitar string plays a note, the air (or water in the
case of underwater concert) vibrate at the that frequency.
As the speed is different in string and air, wavelengths are
too.
 For the given medium low and high freq have the same
speed – higher freq waves have smaller wavelength, and
lower freq waves have longer wavelengths since product
λf = v is same. Concert – all frequencies played at the
same time will reach your ear simultaneously because
they have the same speed. MUSIC.
The Doppler Effect
is an apparent (observed)
change in frequency and
wavelength of a wave
occurring when the source
and observer are in motion
relative to each other, with
the observed frequency
increasing when the source and observer approach
each other and decreasing when they move apart.
 If a source of sound is moving toward you at constant
speed, you hear a higher freq than when it is at rest
 If it is moving at increasing speed you hear higher and
higher freq
 If a source of sound is moving away from you, you hear a
lower freq than when it is at rest
 If it is moving at increasing speed you hear lower and
lower freq
 The same effect occurs with light waves and radar waves
Applications
The Doppler effect is the basis of a technique used to
measure the speed of flow of blood.
Ultrasound (highfrequency sound
waves) are directed
into an artery. The
waves are reflected by
blood cells back to a
receiver. The
frequency detected at
the receiver fr relative
to that emitted by the source f indicates the cell’s speed and
the speed of the blood.
A similar arrangement is used to measure the speed of
cars, but microwaves (EM waves) are used instead of
ultrasound.
When radar waves reflect from a moving object (echo ) the
frequency of the reflected wave changes by an amount that
depends on how fast the object is moving. The detector
senses the frequency shift and translates this into a speed.
Doppler effect is the characteristic of EM waves too.
Based on calculations using the Doppler effect, it appears
that nearby galaxies are moving away from us at speed of
about 250,000 m/s. The distant galaxies are moving away at
speeds up to 90 percent the speed of light. The universe is
moving apart and expanding in all directions.
Astronomy: the velocities of distant galaxies can be
determined from the Doppler shift.
If a star had a fixed position relative to the earth, the light of
one particular frequency would look like this:
5
As the star moves toward us the observed frequency
increases, we say it shifts toward the higher frequency .
That’s why we call it a blue shift.
Law of reflection
The incident and reflected wavefronts.
As the star moves away from us the observed frequency
decreases, we say it shifts toward the lower frequency.
That’s why we call it a red shift.
Red light has the lowest frequency and blue has the highest
out of all of the visible lights.
The EM spectrum coming from stars/galaxies is compared to
one obtained in the laboratory emitted from same elements
(He, H), or to one coming from our Sun.
Most distant galaxies are observed to be red-shifted in the
color of their light, which indicates that they are moving away
from the Earth. Some galaxies, however, are moving toward
us, and their light shows a blue shift.
Edwin Hubble discovered the red shift in the 1920's.
His discovery led to him formulating the Big Bang Theory of
the Universe's origin.
Reflection and refraction of waves
Angle of reflection is equal to angle of incidence.
(the angles are measured to the normal to the interface).
All waves, including light, sound, water obey this relationship,
the law of reflection.
Refraction
When a wave passes from one medium to another, its
velocity changes. The change in speed results in a
change in direction of propagation of the refracted wave.
Visualization of refraction
We now look to see what happens to a wave when it is
incident on the boundary between two media.
When a wave strikes a
boundary between two
media some of it is
reflected, some is
absorbed and some of it
is transmitted.
How much of each?
That depends on the
media and the wave
itself..
Reflection of waves
As a toy car rolls from a
hardwood floor onto carpet, it
changes direction because the
wheel that hits the carpet first
is slowed down first.
The incident and refracted wavefronts.
All waves can be reflected.
First of all, we shall look at a single pulse travelling along a
string.
The end of the rope if fixed – reflected pulse returns inverted.
Free end – reflected pulse is not inverted.
f=
v1
v
= 2
λ1
λ2
frequency is determined by the source so it
doesn’t change. Only wavelength
changes. Wavelength is smaller in
the medium with smaller speed.
6
A mathematical law which will tell us exactly HOW MUCH
the direction has changed is called SNELL'S LAW.
 Where is the fish? Deeper than you think!
Although it can be derived by using little geometry and
algebra, it was introduced as experimental law for light in
1621.
Law of refraction – Snell’s law
For a given pair of
media, the ratio
sin θ1
v
= 1
sin θ2
v2
 Where is the ball? Closer than you think!
is constant for the
given frequency.
The Snell’s law is of course valid for all types of waves.
greater v → greater angle
The speed of light inside matter
 The speed of light in vacuum is:
c = 300,000,000 m/s = 3 x 108 m/s
 In any other medium such as water or glass, light travels
at a lower speed.
Dispersion
 As c is greater than v for all media, n will always be > 1.
Even though all colors of the
visible spectrum travel with the
same speed in vacuum, the
speed of the colors of the
visible spectrum varies when
they pass through a
transparent medium like glass
and water. That is, the refractive index of glass is different for
different colours.
Different colors are refracted by different amounts.
 greater n – smaller speed of light.
Total internal reflection
 INDEX OF REFRACTION, n, of the medium is the ratio of
the speed of light in a vacuum, c, and the speed of light, v,
in that medium:
n=
c
v
no units
 As the speed of light in air is almost equal to c, nair ~ 1
 Refraction of light
The refracted ray is refracted more in the medium
with greater n / slower speed of light
Angle of refraction is greater than angle of incidence. As the
angle of incidence increases, so does angle of refraction.
The intensity of refracted light decreases, intensity of
reflected light increases until angle of incidence is such that
angle of refraction is 900.
Critical angle: θc - angle of incidence for which angle of
refraction is 900
7
Diffraction
Interference - Superposition
When waves pass through a small opening, or pass the edge
of a obstacle, they always spread out to some extent into the
region that is not directly in the path of the waves - into the
region of the geometrical shadow
 Property that distinguishes
waves from particles: waves
can superpose (when
overlapping) and as the
result a lot of possible
craziness can happen. After
two waves overlap they carry
on with exactly the same
properties as before, as if nothing had happened.
The spreading of a wave into a region behind an
obstruction is called diffraction.
Diffraction effects depend on λ of the waves compared
to the size of the opening or an object in the path of the
waves.
Principle of superposition y = y1 + y2
When two or more waves overlap, the resultant
displacement at any point and at any instant is the sum
of the displacements of the individual waves at that
point.
constructive interference –
increased amplitude,
increased energy (E ~ A2 ) –
Small diffraction by a large
opening
Small diffraction around
large object
increased intensity – brighter light
or loud sound at point
the waves are in phase
The waves are diffracted
more through a narrow
opening, when wavelength
is larger than the opening.
Strong diffraction around
small object.
destructive interference –
decreased amplitude,
decreased energy – decreased
intensity –
no light or no sound
the waves are out of phase
remember: big wavelength - big diffraction effects
partially
destructive
interference.
For example, if two rooms are connected by an
open doorway and a sound is produced in a
remote corner of one of them, a person in the
other room will hear the sound as if it originated at
the doorway.
Diffraction provides the
reason
why we can hear
something even if we can
not see it (light waves
have very small
wavelength, so they do not
diffract around big object).
Ultrasound (f > 20 kHz, λ < 1.7 cm)
is used for echolocation: dolphins, bats, sonar.
But why ultrasound? Because of diffraction!!! Or should
we say because of no diffraction!!!
Low frequency sound has longer wavelength, so they will be
diffracted, so not being able to detect the prey. High
frequency sound has smaller wavelength, so it will be
reflected back from the prey. That’s how a bat “sees” its
prey.
Standing waves
are the result of the interference of two identical waves with
the same frequency and the same amplitude traveling in
opposite direction.
A antinode is a point where the standing wave has maximal
amplitude
A node is a point where the standing wave has minimal
amplitude
Distance between two nodes is λ/2
8
only standing wave that
has wavelength
𝜆𝑛 =
2𝐿
𝑛
can be formed on the
string of length L.
v
Wave with
1
= 2L has freq. 𝑓1 =
λ1
=
Wave with
2
= L has freq.
𝑓2 =
λ2
Wave with
n
= 2L/n has freq. 𝑓𝑛 =
v
=
v
𝜆𝑛
v
2L
v
L
…
=𝑛
v
2L
= 𝑛 𝑓1
The frequencies at which standing waves are produced are
called natural frequencies or resonant frequencies of the
string or pipe or...
the lowest freq. standing wave is called FUNDAMENTAL or
the FIRST HARMONICS
The higher freq. standing waves are called HARMONICS
(second, third...) or OVERTONES
Beats
are a periodic variation in loudness (amplitude) – throbbing due to interference of two tones of slightly different
frequency.
Two waves with
slightly different
frequencies are
travelling to the
right. The resulting
wave travels in the
same direction and with the same speed as the two
component waves.
The beat frequency is equal to the absolute value of
the difference in frequencies of the two waves.
The beat frequency is equal to the absolute value of
the difference in frequencies of the two waves.
𝑓 = |𝑓1 − 𝑓2 |
9
Examples:
1. A sound wave produced by a clock chime is heard 515 m
away 1.5 s later.
(a) What is the speed of sound in the air there?
(b) The sound wave has a frequency of 436 Hz. What is the
period of the wave?
(c) What is the wave's wavelength?
(a) v = d/t = 515/1.5 = 343 m/s
(b) f = 1/T = 1/436 = 2.29x10-3 s
(c) v =  f →  = v / f = 0.87 m
2. A hiker shouts toward a vertical cliff 465 m away. The echo
is heard 2.75 s later.
(a) What is the speed of sound in air there?
(b) The wavelength of the sound is 0.75 m. What is the
frequency of the wave?
(a) v = distance/time = 2d/t = 2·465/2.75 = 338 m/s
(c) What is its period?
(b) v =  f → f = v/ = 338/0.75 = 451 Hz
(c) T = 1/f = 2.22x10-3 s
If you wanted to increase the wavelength of waves in a rope
should you shake it at a higher or lower frequency?
v=f
v depends only on the medium. Therefore for given medium
it is constant. So if wavelength increases the frequency
decreases. You should shake it at lower frequencies. CHECK
IT, PLEASE
3. A stone is thrown onto a still water surface and creates a
wave. A small floating cork 1.0 m away from the impact point
has the following displacement—time graph (time is
measured from the instant the stone hits the water):
Find
(a) amplitude
(b) the speed of the wave
(c) the freq.
(d) wavelength
(a)
(b)
(c)
(d)
A = 2 cm
v = d/t = 1/1.5 = 0.67 m/s
f = 1/T = 1/0.3 = 3.33 Hz
λ = v/f = 0.666/3.333 = 0.2 m
Example: To echolocate an object one must have both
emitter and detector.
If the wavelength of an emitted wave is smaller than the
obstacle which it encounters, the wave is not able to diffract
around the obstacle, instead the wave reflects off the
obstacle. Reflected wave is caught by detector giving it
information on how far (2d = vt) and how big is the object
(reflection from different directions)
The ultrasound bats typically chirp is ~ 50 000 Hz.
What is wavelength of that sound?
The speed of sound wave in air is ~ 340 m/s.
v = λf
so λ = v/f
λ = 0.0068 m = 0.7 cm
So, bats use ultrasonic waves with λ smaller than the
dimensions of their prey (moth – couple of centimeters).
Example: (a) Calculate the wavelengths of
i) FM radio waves of frequency 96 MHz.
λ = c/f = 3x108/96x106 = 3.1 m
ii) AM long wave radio waves of frequency 200 kHz.
λ = c/f = 3x108/200x103 = 1500 m
(b) Use your answers to (i) and (ii) to explain why if your
car is tuned to FM, it cuts out when you enter a tunnel
but doesn’t if you are tuned to long wave reception.
Lower-frequency (longer-wavelength) waves can
diffract around larger obstacles, while high-frequency
waves are simply stopped by the same obstacles.
This is why AM (~1 MHz, 300 m wavelength) signals
can diffract around a building, still producing a usable
signal on the other side, while FM (~100 MHz, 3 m
wavelength) signals essentially require a line-of-sight
path between transmitter and receiver.
Example: Suggest one reason why ships at sea use a very
low frequency sound for their foghorn.
Low frequency sounds do propagate much further
than high-frequency ones – due to diffraction.
Another explanation is that the method of generating
the sound involves the production of a very strong
pressure pulse. The fog horn is loud so that it can be
heard far away.
Elephants also use these deep sounds to
communicate over long distance.