●
Interference occurs whenever two or
more waves overlap.
in physics,
the net effect of the combination
of two or more wave trains
moving on intersecting or
coincident paths. The effect is
that of the addition of the
amplitudes of the individual
waves at each point affected
by more than one wave.
1
17.1 The Principle of Linear Superposition
When the pulses merge, the Slinky
assumes a shape that is the sum of
the shapes of the individual pulses.
17.1 The Principle of Linear Superposition
When the pulses merge, the Slinky
assumes a shape that is the sum of
the shapes of the individual pulses.
17.1 The Principle of Linear Superposition
THE PRINCIPLE OF LINEAR SUPERPOSITION
When two or more waves are present simultaneously at the same place,
the resultant disturbance is the sum of the disturbances from the individual
waves.
5
Interference of sound. In a concert room of bad quality, you can't
Hear anything at some places. Depending if the waves
Arrive in phase (constructive interference, the crests add up) or
They arrive our of phase (destructive interference, crest + through )
You can hear a
sound
You can't hear a
sound
6
17.2 Constructive and Destructive Interference of Sound Waves
When two waves always meet condensation-to-condensation and
rarefaction-to-rarefaction, they are said to be exactly in phase and
to exhibit constructive interference.
17.2 Constructive and Destructive Interference of Sound Waves
When two waves always meet condensation-to-rarefaction, they are
said to be exactly out of phase and to exhibit destructive interference.
When they are in phase, one WAVE travels 1,2,3 … wavelengths farther
Than the other When they are out of phase the difference
In distance they waves travel is 0.5, 1.5, 2.5 .... wavelengths
Examples:
1)
A person stands in front of 2 speakers that are emitting the same
Pure tone. The person moves to one side until no sound is
Hears. At that point , the person is 7m from one speakers and
7.2 meter from the other. What is the frequency of the tone being
Emitted?
2)
2 speakers producing identical sound in phase in an open field , feed into
A microphone can detect no sound when it is 4.7 m from one speaker
And 5.2 m from the other, what is the wavelength of the sound ?
9
17.2 Constructive and Destructive Interference of Sound Waves
Example 1 What Does a Listener Hear?
Two in-phase loudspeakers, A and B, are
separated by 3.20 m. A listener is stationed
at C, which is 2.40 m in front of speaker B.
Both speakers are playing identical 214-Hz
tones, and the speed of sound is 343 m/s.
Does the listener hear a loud sound, or no sound?
17.2 Constructive and Destructive Interference of Sound Waves
Calculate the path length difference.
( 3.20 m ) 2 + ( 2.40 m ) 2 − 2.40 m = 1.60 m
Calculate the wavelength.
λ =
v 343 m s
=
= 1.60 m
f
214 Hz
Because the path length difference is equal to an integer (1)
number of wavelengths, there is constructive interference, which
means there is a loud sound.
APPLICATION:MNOISE CANCELING HEAPHONES
UTILIZES INTERFERENCE.
12
BEATS
http://paws.kettering.edu/~drussell/Demos/superposition/superposition.html
When we add 2 waves with a small difference in frequency, we get beats.
See applets
http://www.youtube.com/watch?v=dD9gtq08tss&feature=related
13
17.4 Beats
The beat frequency is the difference between the two sound
frequencies.
Examples:
1) A piano tuner adjusts a string to sound C-264Hz. In trying to match
A second a second string to it, he sounds both together and hears
Three beats per second. (3 cycles per second).
What is the frequency of the other string,
Hint: the frequency of the beats = difference in frequencies of the 2 waves
Being added.
2) A piano tuner uses a tuning fork to adjust the key that plays the A note
Above middle C. (whose frequency is 440Hz). The tuning fork emits
A perfect 440Hz tone. Then the tuning fork and the piano are struck
Beats of frequency 3 Hz are heard. What is the frequency of the piano key ?
(2 choices)
3) If its known that the piano is too high, should the piano tuner tighten or loosen
The wire inside the piano to tune it?
Another phonomenon due to interference of waves: diffraction
Diffraction of waves
the spreading of waves around
obstacles.
Diffraction takes place with sound;
with electromagnetic radiation,
such as light, X-rays, and
gamma rays; and with very smal
moving particles such as atoms,
neutrons, and electrons,
which show wavelike properties.
16
Source: encyclopedia britannica
●
●
●
Diffraction results whenever a wave
has to travel past a barrier or
obstruction.
As the wave travels through the
opening, the outgoing waves bend.
The amount of diffraction
depends on the wavelength
and the size of the
obstruction.
http://www.youtube.com/watch?v=4EDr2YY9lyA
You can hear behind
17
a door because sound is
Diffracted. The wavelength
Of sound is greater than
door opening.
Aspects of wave
propagation, cont’d
●
●
When the opening is much larger than
the wavelength, there is little diffraction.
The amount of diffraction increases as
the wavelength
becomes more
similar to the
size of the
opening.
18
See applet
17.3
Diffraction
The bending of a wave around
an obstacle or the edges of an
opening is called diffraction.
17.3
Diffraction
single slit – first minimum
λ
sin θ =
D
17.3
Diffraction
Circular opening – first minimum
λ
sin θ = 1.22
D
17.3
Diffraction
A 1500 Hz sound and a 8500Hz each
Emerges from a loudspeaker through a circular
Opening that has a diameter of 0.3m.
Assuming that the speed of sound is 343m/s,
Find the diffraction angle for each sound.
Circular opening – first minimum
λ
sin θ = 1.22
D
So the wavelength of the
Wave has to be larger
Than the opening for diffraction
to happen.
Light has a very small wavelength
10-7 me so the opening has to be really
Small to see the diffraction.,
The longer the wave compared to the obstruction, the greater the diffraction
is. Long waves are better at filling shadows. (see above).
This is why foghorns operate at long frequency
sound waves. To fill blind spots.
Likewise AM radio waves are long compared to buildings and objects
On their way. Long waves don't “ see” small building in their path.
FM radio waves aren't received as well as they tend to diffract more.
TV use the same range of wavelengths as AM radio. That's why antennas
Has to be placed on top of buildings.
23
27.5 Diffraction also applied to light waves.
Diffraction helps when broadcasting with AM radio waves but
Is a problem when using microscopes.
If the wavelength of light is larger than the size of the object
We ant to observe, diffraction makes the image blurry and if
The object is too small we can't see anything. What is the solution ?
Instead of using visible light, use smaller wavelength.
http://physics2life.blogspot.com/2008_10_01_archive.html
25
Wavelengths of visible light
1 nanometer = 10-9 so visible light between 4 10-7m and 7 10-7 m
Between .4 and 0.7 micrometer. So the object to be seen has
To be larger than these values. Above 1 micrometer = millionth of a meter
1 thin hair is 25 micro. Can you see it ?
1 red cell is 8 micro. Can you see it ?
1 strand of DNA (thickness ) is 0.01 micro .
Can you see it ?
26
The electron microscopes uses electron with smaller wavelength
Than visible light to “ see” small objects. (electrons are waves).
Bacteria x 2 million
Wavelengths are between:
1 nm – 1 μm
0.001 – 1 micro
microorganism
27
Pollen grains
with electrons
Using photons with high energy we can
See large molecules.
28
29
In 2009 IBM developed a method to probe even smaller structures.
They use an atomic needle that probes the molecules and atoms.
atomic force microscope Here how it works:
http://scienceblogs.com/startswithabang/2009/08/just_a_quick_little_picture.php
pentacene molecule
1.4 nanometer across !
You see the electronic cloud.
Using this technology
You can move one atom
At a time.
Size on atom = 10-10m
0.1 nm
30
17.5 Transverse Standing Waves
On a string, a 2 traveling waves interfere with each other
To produce: Transverse standing wave patterns.
17.5 Transverse Standing Waves
In reflecting from the wall, a
forward-traveling half-cycle
becomes a backward-traveling
half-cycle that is inverted.
Unless the timing is right, the
newly formed and reflected cycles
tend to offset one another.
Repeated reinforcement between
newly created and reflected cycles
causes a large amplitude standing
wave to develop.
Standing waves/interference
standing wave (Encyclopædia Britannica)
combination of two waves moving in opposite directions, each having the same
amplitude and frequency. The phenomenon is the result of interference—that is,
when waves are superimposed, their energies are either added together or canceled
out.
In the case of waves moving in the same direction, interference produces a traveling
wave; for oppositely moving waves, interference produces an oscillating wave fixed in
space.
Check this applet. The green is the coming wave produced by the source, the
Blue is the reflected one. They add up to form a standing wave with nodes
And antinodes.
http://www.phys.unsw.edu.au/jw/strings.html
33
Standing waves/ Harmonics
Waves can be at the same place and at the same time and they interfere.
When a wave reflects inside a medium ( a string, flute, bridge..), it interferes with
Itself producing constructive or destructive interference patterns.
These patterns are called standing waves or harmonics.
Standing wave=a stationary wave pattern formed in a medium when 2 sets
Of identical waves pass through the medium in opposite direction.
Here how to produce harmonics on A string: (see first applet)
http://www.physics.uiowa.edu/~umallik/adventure/music04.htm
http://www.walter-fendt.de/ph14e/stwaverefl.htm
For a given length of a string, only waves with certain frequencies can create
standing waves. They other waves don't “ stand”, they die.
This is because the distance from one node to the next must always be some
fraction of the total length. These harmonics correspond to the natural frequencies
Of the string ot mode of vibration. See also:
http://faraday.physics.utoronto.ca/IYearLab/Intros/StandingWaves/Flash/reflect.html
Making harmonics with fire :
http://www.youtube.com/watch?v=UR8rZMKlIq4
34
In 2D:
http://www.youtube.com/watch?v=tliBfYdddhU
17.5 Transverse Standing
Waves. Find the wavelengths
http://www.youtube.com/watch?v=3BN5-JSsu_4&feature=related
String fixed at both ends
 v 
f n = n

 2L 
n = 1, 2, 3, 4, 
These are the frequencies that generate the standing waves.
These are the harmonic (or resonant) frequencies. The first standing wave,
The one for which the harmonic number is 1 (n=1) , is called the the fundamental
Standing wave. The nth harmonic frequency is n times the fundamental frequency.
f n = nf o
 2L 
λn = 

 n 
n = 1, 2, 3, 4,
Examples:
2) A string of length 12 m that's fixed at both ends supports a standing wave
With a total of 5 nodes. What are the harmonic number and wavelengths
Of this standing wave ?
1) What is the fundamental frequency sounded by a violin string 45cm
Long if the wave in the string is 280m/s ?
3) what are the wavelengths of the four longest waves that can stand on a
String 60cm long ?
4) what are the four longest frequencies
Of a 60cm string if the speed
Of the wave is 240m/s ?
 2L 
λn = 

 n 
 v 
f n = n

 2L 
N=4
n = 1, 2, 3, 4,
n = 1, 2, 3, 4, 
If L = 1m,
Find the frequencies.
v=208m/s
17.5 Transverse Standing Waves
 v 
f n = n

 2L 
n = 1, 2, 3, 4, 
If you decrease the length of the string, you increase the frequencies.
The heaviest string on an electric guitar has a linear density of 5.28 10-3 km/m
And is stretched with a tension of 226N. This string produces the musical
Note E when vibrating along its entire length in standing wave at the
Fundamental frequency 164.8 Hz .
A) Find the length of the string between its 2 fixed ends.
B) A player wants the string to vibrate at 2x164.8 Hz= 329.6 Hz
As it must be if the musical note E is to be sounded one octave higher in pitch.
To accomplish this, he presses the string against the proper fret before
Plucking the string. Find the distance L between the fret and the bridge
Of the guitar.
http://zonalandeducation.com/mstm/physics/waves/standingWaves/standingWaves1/StandingWaves1.html
 v 
f n = n

2
L


n = 1, 2, 3, 4, 
Creating standing waves with the 2 ends free. Like in a flute.
Sound waves reinforce each other only for given frequencies.
All the overtones are in a whole number ratio of the fundamental.
 2L 
λn = 

 n 
 v 
f n = n

 2L 
n = 1, 2, 3, 4,
n = 1, 2, 3, 4,41

http://zonalandeducation.com/mstm/physics/waves/standingWaves/standingWaves2/StandingWaves2.html
17.6 Longitudinal Standing Waves
Tube open at both ends
 v 
f n = n

 2L 
n = 1, 2, 3, 4, 
17.6 Longitudinal Standing Waves
Example 6 Playing a Flute
When all the holes are closed on one type of
flute, the lowest note it can sound is middle
C (261.6 Hz). If the speed of sound is 343 m/s,
and the flute is assumed to be a cylinder open
at both ends, determine the distance L.
17.6 Longitudinal Standing Waves
 v 
f n = n

 2L 
n = 1, 2, 3, 4, 
nv
1( 343 m s )
L=
=
= 0.656 m
2 f n 2( 261.6 Hz )
Example: With the temperature at 15C, what are the three lowest frequencies
Sounded by an open organ pipe 40 cm long ?
You can also creates harmonics by blowing in a organ pipe or a bottle.
In that case you are creating sound waves that interfere with each
Other in a constructive or destructive way. Antinode = high pressure
(compression) node= low pressure .
f=fo
fundamental
f=2fo
f=3fo
 v 
f n = n

 4L 
 4L 
λn = 

 n 
n = 1, 3, 5,
n = 1, 3, 5,
45
http://faraday.physics.utoronto.ca/IYearLab/Intros/StandingWaves/Flash/sta1fix.html
http://zonalandeducation.com/mstm/physics/waves/standingWaves/standingWaves3/StandingWaves3.html
17.6 Longitudinal Standing Waves
Tube open at one end
 v 
f n = n

 4L 
n = 1, 3, 5,
Examples:
1) What are the fundamental and first overtone frequencies of a closed
Organ pipe 35cm long when the temperature is 18 C ?
2) A closed=end tube resonate at a fundamental frequency of 343 Hz.
The air in the tube is at a temperature of 20C, and it conducts sound
At a speed of 343m/s
A) what is the length of the tubes ?
B) What is the next higher harmonic frequency ?
C) answer the questions if you suppose the tube was open at its far end,
 v 
f n = n

 4L 
 v 
f n = n

 2L 
n = 1, 3, 5,
n = 1, 2, 3, 4, 
resonnance:
Natural frequency=frequency at which an elastic object naturally
Tends to vibrate, so that minimum energy is required to produce a forced
Vibration or to continue vibration at that frequency.
The harmonics happen at the natural frequencies.
If you force an object to vibrate and if the frequency the force is the same
As the natural frequency, then the object resonate. The displaced
Molecules reach larger and larger amplitude. This can have dramatic
Consequence: if soldiers march across a bridge and if the frequency of
Their steppings matches the natural frequency of the bridge, it will vibrate
With higher and higher amplitude until it collapses.
Resonance= The response of a body when a forcing frequency matches its
Natural frequency
http://www.youtube.com/watch?v=17tqXgvCN0E
fun:
http://www.youtube.com/watch?v=pz1j-GSSitg&NR=1
http://www.youtube.com/watch?v=j-zczJXSxnw&feature=related
The case of a violin and guitar resonate.
48
To see animation
See original
slides
http://www.youtube.com/watch?v=zWKiWaiM3Pw&feature=related
When you tune your favorite radio, the antenna is set to resonate
With a radio wave of a given frequency.
resonnance:
in physics, relatively large selective response of an object or a system that vibrates
in step or phase, with an externally applied oscillatory force. Resonance was first
investigated in acoustical systems such as musical instruments and the human voice.
49
An example of acoustical resonance is the vibration induced in a violin or piano string
of a given pitch when a musical note of the same pitch is sung or played nearby.
Encyclopedia britannica
Review for the test:
1) In a longitudinal wave the distance between a compression to the nearest rarefaction is 35 cm. If the frequency
Of the wave is 4Hz , how fast is the wave traveling ?
2) The the temperature at 10C , 2 loudspeaker are producing a 680Hz tone in phase with each other.
If you locate 1 node at 15.8m from the nearer speaker, how far are you from the the other one ?
3) you are tuning a 12-string guitar and adjust one string to sound 220Hz. When you strike this along with its
matching matching string you hear beats of 4Hz. You find if you tighten the matching string string a little
The beat disappear . What was the frequency of the matching string ?
4) A rope that transmits a wave at 8m/s at one end at a frequency of 6Hz. If the other end is fastened down, how
far from that end are the first antinodes ?
5) Determine the fundamental and the first 2 overtones of a violin string 35cm long if the speed of the wave is
180m/s.
6) Determine the fundamental and its first overtone of a tuning fork whose tines are 18cm long. If the speed
Of the wave is 60m/s.
7) Determine the fundamental and the first overtone of a closed organ pipe 35cm long at a temperature of 20C
8) fill (read the music part of the slides first)
1) the pitch of a musical sound depends on the _____________ of the wave
2) the frequency of a note 2 octaves below A440Hz is _______________
3) overtones determine the ___________ of a musical sound
4) diffraction around corners is greater when the ____________ is larger.
5) destructive interference occurs when 2 waves arrive at a point in ________ with each other.
6) beat frequency is the _____________ of the frequencies of 2 waves.
7) In a standing wave the distance between 2 antinodes is the _____________
50
Fourier theorem:
Any complex wave with a periodic pattern can be decomposed into the sum of
Sines for cosines. Below green + blue = red
51
Waves — types and
properties, cont’d
●
A complex wave is any continuous
wave that does not have a sinusoidal
shape.
Any complex wave with a periodic pattern can be decomposed into the sum of
Sines for cosines. Fourier discovered this principle.
52
●
A waveform of a sound
wave is a graph of the
air-pressure fluctuations
causes by the sound
wave versus time.
A pure tone is a sound
with a sinusoidal
waveform.
 A complex wave is a
sound that is not pure.

Wavelength λ and f=v/λ
Find pitch if green = 3m, v=344m/s
music
noise
53
The graph representing a musical sound has a shape that repeats itself over
And over again. The graph has a given frequency called the pitch.
The graph representing a musical sound has a shape that repeats itself over
And over again. The graph has a given frequency.
The frequency of the waveform define the pitch of the sound.
The middle A has a pitch of 440 Hz. The next A has a pitch of 880Hz
The interval is called an octave. The frequency of the waveform double.
54
Source http://amath.colorado.edu/pub/matlab/music/
Waveforms of
Sound waves for
A pitch of :
440Hz, 880Hz,1760Hz
You can hear them here:
1 octave =
Double the frequency.
http://amath.colorado.edu/pub/matlab/music/
55
The middle A has a pitch of 440 Hz. Why this note does not sound the
Same when played by different instruments ?
In fact the waveform corresponding to a pitch of 440Hz (middleA)
Is not the same for all the instruments. The waveform is more complex for a
violin than for a flute for example.
Waveforms of middle A
For different instruments, pitch is
The same but not the tone quality.
You can hear them here:
http://amath.colorado.edu/pub/matlab/music/
Because the waveform is not the
Same, our ears is not going
To perceive the same tone quality
Or timbre.
56
●
Tone quality is a measure by which
two sounds of the same frequency and
amplitude sound different.

●
A sax sounds different from a trumpet
playing the same note because the two
instruments have different tone qualities.
We typically refer to tone quality as
timbre or tone color.
57
Notes sound harmonious when their frequencies are in simpler whole number
Ratios. The diatonic scale is built on such ratios.
The group of 3 notes called a major triad such as C-E.G
Or F-A-C has its frequencies in the ration 4:5:6.
Diatonic scale:
Middle A = 440Hz, B=495Hz, C=528Hz, E=660Hz, F=704Hz, G=792Hz, A=880Hz
1)With a temperature at 22C, a violin plays a middle C at 264 Hz. What is the
Wavelength of the second overtone of the sounds wave produced in air ?
2)If a major triad is formed by the notes A – C# - E , what is the frequency of C# ?
3) What is the frequency of the A 2 octaves below A440Hz ?
58
So Why the waveforms are different for different notes and
Different instruments? We are back to Fourier principle
Any complex waveform is the weighted sum of pure tones
Or harmonics.
Which harmonics you are adding and they amplitude of each (weight)
Makes up the quality of the tone or timbre.
http://mysite.verizon.net/vzeoacw1/harmonics.html
http://www.howmusicworks.org/hmw104.html
59
Frequency spectrum when a guitar
Is plucked in the usual position
Musical sounds are composed
Of many frequencies
Called partial tones
The lowest frequency of a
Musical note is called
The fundamental.
The fundamental determines
The pitch of the note.
Partial tones are whole multiples of the fundamental and are called harmonics.
A tone that has twice the frequency of the fundamental is called
The 2nd harmonic … (for a string instrument or a flute)
The frequencies of the harmonics
are whole-numbered multiples
of the complex waveform’s frequency.
60
All these instruments below are playing the same note, with
The same pitch, the same frequency of the waveform, the same
Fundamental (1st harmonic) but the waveform is not the same.
The amplitude and the number of the overtones are not the same.
middleA
61
Source: http://amath.colorado.edu/pub/matlab/music/
Piano. Here the pitch doubles (fundamental x 2). The harmonic spectrum
Changes too.
http://www.youtube.com/watch?v=CGgpEGRkzLQ&feature=related
http://www.youtube.com/watch?v=121DoSs62eY
62
●
The specific tone quality of a sound
depends on:
the number of harmonics that are present,
and
 the relative amplitudes of these harmonics

●
A spectrum analyzer displays a
complex waveform in terms of the
constituent harmonics.
63
Equal temperament scale: (piano)
Note
C (do)
C# (Db)
D (re)
D# (Eb)
E (mi)
F (fa)
F# (Gb)
G (sol)
G# (Ab)
A (la)
A# (Bb)
B (si)
C(do)
f (Hz)
Ratios
261.6 1.05946 = 12th root of 2
277.2 1.05946
293.7
1.05946
Galileo deveoped the western music scale.
311.1 1.05946
There is a ratio of 21/12
329.6
1.05946
(12th root of 2) between 2 consecutive
349.2
1.05946
notes.
370 1.05946
392
1.05946
415.3
1.05946
440.0 1.05946
466.2
1.05946
493.9
1.05946
523.3
More
http://www.phy.mtu.edu/~suits/scales.html
64
Just tuning scale : used for ensembles
overtone series for simple systems such as vibrating strings or air columns
2 notes sound good together if their frequencies (fundamental)
Are in a whole number ratio.
Between C and the higher C there is a ratio of 2:1 = called an octave.
Frequency ratios (same as for the previous scale)
Note
f (Hz)
C (do)
264.0
D (re)
297.0
E (mi)
330.0
F (fa)
352.0
G (sol)
396.0
A (la)
440.0
B (si)
495.0
C(do)
528.0
5/4
9/8
4/3
A fourth
4/3
3/2
6/5
A perfect fifth
A fourth
5/4
4/3
5/4
6/5
A fourth
65
Additional stuff :
Production of sound
●
Sound can be produced by:

Causing a body to vibrate:


Varying an air flow:


e.g., buzzing your lips
Abrupt changes in an object’s temperature:


e.g., plucking a string
e.g., a lightning flash creates thunder
By creating a shock wave:

e.g., flying faster than the speed of sound
66
Production of sound
●
A piano produces sound by:



The player presses a key so that the hammer strikes
the wire.
The wire vibrates
and transmits this
vibration to the
soundboard.
The soundboard
then radiates the
sound to the room.
67
●
In a room, we have to deal with the
multiple reflections off the walls and
other objects.

●
It gets a lot more complicated to determine
the amplitude at an arbitrary place in the
room.
The process of repeated reflections of
sound in an enclosure is called
reverberation.
68
Propagation of sound, cont’d
●
●
A hand clap in an open field is a simple
pulse since there is no echo.
But in a gym, there
are multiple echoes
which tends to
make the sound
fade away
gradually.
69