Making Waves
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A wave is a disturbance that travels through something –
solids, liquids or gases
The disturbance moves because of the elastic nature of the
material
As the disturbance moves, the parts of the material
(segment of string, air molecules) execute harmonic motion
(move up and down or back and forth)
The “wave” - people stand up then sit down, then the
people next to them do the same until the standing and
sitting goes all around the stadium.
the standing and sitting is the disturbance
notice that the people move up and down but the
disturbance goes sideways !
Physics 101: Chapter 15, Pg 1
Why are waves important?
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Î waves carry energy Í
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they provide a means to transport energy from one place
to another
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the energy from the sun comes to us along
electromagnetic waves– light waves
Physics 101: Chapter 15, Pg 2
Types of waves
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There are three types of waves:
Mechanical waves require a material medium to travel (air, water,
ropes). These waves are divided into three different types.
ÍTransverse waves cause the medium to move perpendicular to
the direction of the wave.
ÍLongitudinal waves cause the medium to move parallel to the
direction of the wave.
ÍSurface waves are both transverse waves and longitudinal
waves mixed in one medium.
Electromagnetic waves do not require a medium to travel (light,
radio).
Matter waves are produced by electrons and particles.
Physics 101: Chapter 15, Pg 3
Mechanical waves
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a disturbance that propagates through a medium
» waves on strings
» waves in water
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ocean waves
ripples that move outward when a stone is thrown in a
pond
» sound waves – pressure waves in air
Physics 101: Chapter 15, Pg 4
Types of Waves -- Transverse
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In a transverse wave, each element that is disturbed moves
perpendicularly to the wave motion
Physics 101: Chapter 15, Pg 5
Types of Waves -- Longitudinal
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In a longitudinal wave, the elements of the medium undergo
displacements parallel to the motion of the wave
Physics 101: Chapter 15, Pg 6
Some terminology
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the maximum displacement of an object from
equilibrium is called the AMPLITUDE A
the time that it takes to complete one full cycle is called
the PERIOD T of the motion
if we count the number of full cycles the oscillator
completes in a given time, that is called the
FREQUENCY f of the oscillator
frequency f = 1 / period = 1 / T
The wavelength = wave speed / frequency
» = v / f or Î v = λ × f
The speed of the wave: v
Physics 101: Chapter 15, Pg 7
Example: wave on a string
2 cm
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2 cm
2 cm
A wave moves on a string at a speed of 4 cm/s
A snapshot of the motion reveals that the wavelength(λ) is 2
cm, what is the frequency (ƒ)?
v = λ׃, so ƒ = v ÷ λ = (4 cm/s ) / (2 cm) = 2 Hz
Physics 101: Chapter 15, Pg 8
How fast does it go?
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The speed of the wave moving to the right is not the same as the
speed of the string moving up and down. (it could be, but that
would be a coincidence!)
The wave speed is determined by:
» the tension in the string
Æ more tension Æ higher speed
» the mass per unit length of the string (whether it’s a heavy
rope or a light rope)
Æ thicker rope Æ lower speed
F
The wave speed on the rope is:
v=
µ
F – the tension in the rope (force)
µ = m/L – unit length
m – total mass of the rope
L – the length of the rope
Physics 101: Chapter 15, Pg 9
Try Box 15.1
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a)
A rope has an overall length of 10 m and a total mass of 2 kg. The
rope is stretched with a tension of 50 N. One and of the rope is
fixed, and the other is moved up and down with frequency of 4 Hz.
Í a) What is the speed of waves on this rope
Í b) what is thee wavelength for the frequency of 4 Hz.
L = 10m, m = 2 kg, F = 50N, v = ?
µ=
b)
m 2kg
=
= 0.2kg / m
L 10m
v=
F
µ
=
50 N
= 15.8m / s
0.2kg / m
f = 4 Hz, λ = ?
λ=
v 15.8m / s
=
= 3.95m
f
4 Hz
Physics 101: Chapter 15, Pg 10
Interference of Waves
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Waves obey the Superposition Principle
ÍIf two or more traveling waves are moving through a
medium, the resulting wave is found by adding together the
displacements of the individual waves point by point
Physics 101: Chapter 15, Pg 11
Constructive Interference
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Two waves, a and b, have
the same frequency and
amplitude
The combined wave, c, has
the same frequency and a
greater amplitude
Physics 101: Chapter 15, Pg 12
Destructive Interference
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Two waves, a and b, have the
same amplitude and frequency
They are 180° out of phase
When they combine, the
waveforms cancel
Physics 101: Chapter 15, Pg 13
Standing Waves
•Sometimes waves appear to be standing still, i.e. the crests and the
troughs appear to stay in the same place. We can see them in water,
especially water surrounded by walls. We call them standing waves
or stationary waves. Musical instruments depend on standing
waves:
•In a string, for example guitar, pianoforte, violoncello.
•In a column of air, e.g. clarinet, tuba, organ.
Stationary waves are formed when two progressive waves are
superposed:
•Equal frequency
•Nearly the same amplitude
•Same speed
•Traveling in opposite directions.
Physics 101: Chapter 15, Pg 14
Standing Waves 2
If we send an incident wave down a string, which is fixed at the end,
the wave is reflected at the fixed end and undergoes a phase change
of p radians or 180o. There is no phase change at the free
end.
Physics 101: Chapter 15, Pg 15
Standing waves 3
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At the NODE positions, the string does not move
At the ANTINODES the string moves up and down
harmonically
Only certain wavelengths can fit into the distance L
The frequency is determined by the velocity and mode
number (wavelength)
Physics 101: Chapter 15, Pg 16
Vibration modes of a string
A
N
N
L
N
A
N
L
N = nodes,
A
N
Fundamental mode
Wavelength = 2 L
Frequency = fo
First harmonic mode
Wavelength = L
Frequency = 2 fo
A = antinodes
Physics 101: Chapter 15, Pg 17
Vibration frequencies
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In general, f = v / λ, where v is the propagation speed of the string
The propagation speed depends on the diameter and tension
Modes
ÍFundamental: fo = v / 2L
ÍFirst harmonic: f1 = v / L = 2 fo
The effective length can be changed by the musician “fingering” the
strings
Physics 101: Chapter 15, Pg 18
Bowed instruments
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In violins, violas, cellos and basses, a bow made of horse hair is used to
excite the strings into vibration
Each of these instruments are successively bigger (longer and heavier
strings).
The shorter strings make the high frequencies and the long strings make
the low frequencies
Bowing excites many vibration modes simultaneouslyÆ mixture of tones
(richness)
Physics 101: Chapter 15, Pg 19
Waves on a Guitar String
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The standing wave on a plucked guitar string has nodes at both ends
The string is fixed at both ends and cannot oscillate at these points
The simplest standing wave is one with nodes at either end and antinodes
in the middle – Top wave on the figure. This wave is called fundamental
wave – first harmonic.
The wavelength of first harmonic is λ = 2L
The wavelength of the interfering wave is determined by the length of the
string – see figure.
LL
L
=
λ
2
L=λ
L =
3
λ
2
Physics 101: Chapter 15, Pg 20
Try Box 15.2.
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a)
A guitar string has a mass of 4 g, a length of 74 cm, and a tension
of 400 N. These values produce a wave speed of 274 m/s.
Í a) what is its fundamental frequency – first harmonic?
Í b) what is the frequency of second harmonic?
L = 74 cm = 0.74 m, v = 274 m/s, λ = 2L, f1 = ?
f1 =
b)
v
λ1
=
v
274m / s
=
= 185Hz
2 L 2 ⋅ 0.74m
λ = L, f2 = ?
f2 =
v
λ2
=
v 274m / s
=
= 370 Hz
L
0.74m
Physics 101: Chapter 15, Pg 21
Practice
A rope has a mass of 2 kg and a length of 10 m. It is stretched
with a tension of 50 N and fixed at both ends. What is the
frequency of the first harmonic on this rope?
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Solution:
Í µ = m/L = 0.2 kg/m, v2 = F/m = 50 N / (0.2 kg/m) = 250 (m/s) 2,
v= 15.8 m/s.
Í The wavelength of the first harmonic is 20 m. The frequency f
is f = v / λ = (15.8 / 20) Hz = 0.79 Hz.
Physics 101: Chapter 15, Pg 22
SOUND WAVES 1
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Any object that vibrates, such as string, an air column, e
membrane, etc can produce sound.
When the vibrating object moves outward, it pushes the air
molecules, creating a region of high pressure.
When it moves inward it creates a region of low pressure.
As the vibrating object alternately compresses and
expands, the surrounding air, the disturbance travels
outward from the source as a longitudinal wave.
Physics 101: Chapter 15, Pg 23
SOUND WAVES 2
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we can only hear sounds between
30 Hz and 20,000 Hz
below 30 Hz is called infrasound
above 20,000 is called ultrasound
Physics 101: Chapter 15, Pg 24
THE SPEED OF SOUND 1
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In room temperature air, sound travels with a speed of 340 m/s.
The factors that determine the speed of sound are related to how
rapidly one molecule transmits changes in velocity to nearby
molecules to propagate the wave.
Sound travels faster in liquids and solids than in gases, since the
particles in liquids and solids are closer together and can respond
more quickly to the motion of their neighbors .
Physics 101: Chapter 15, Pg 25
THE SPEED OF SOUND 2
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Sound waves travel much slower than light waves.
That is why we hear a clap of thunder a few seconds after you
see the flash of lightning.
Since light travels extremely fast, the light flash reaches us
almost instantaneously.
The sound wave takes about 3 seconds to cover 1 km
Counting seconds between the flash and the thunder tells you
how far away the lightning strike happened.
If the flash and the thunder clap occur almost simultaneously,
you may be in trouble.
Physics 101: Chapter 15, Pg 26
STANDING WAVES IN SOUND WAVES 1
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When a sound wave hits a wall, it is partially absorbed and
partially reflected.
A person far enough from the wall will hear the sound twice.
This is an echo.
In a small room the sound is also heard more than once, but
the time differences are so small that the sound just seems to
loom.
Physics 101: Chapter 15, Pg 27
STANDING WAVES IN SOUND WAVES 2
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The standing sound wave on a pipe with one end closed is
analyzed to determine their frequencies.
The longest standing wave in a tube of length L with one open
end and one closed end has a displacement antinodes at the
open end and a displacement node at the closed end. This is
the fundamental.
Physics 101: Chapter 15, Pg 28
STANDING WAVES IN SOUND WAVES 3
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The next longest standing wave in a tube of length in a tube of
length L with one open end and one closed end is the third
harmonic. It also has a displacement antinodes at one end
and a node at the other. .
Physics 101: Chapter 15, Pg 29
DOPPLER EFFECT
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If a source of sound is moving toward you, you hear a higher
frequency than when it is at rest
If a source of sound is moving away from you, you hear a
lower frequency than when it is at rest
You can hear this effect with sirens on fire engines of train
whistles
A similar effect occurs with light waves and radar waves
When radar waves bounce off a moving object (echo )the
frequency of the reflected radar 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.
Physics 101: Chapter 15, Pg 30