Physics 110 Lecture 33 from Chapter 14 Sections 8 to 12

1
Physics 110 Lecture 33 from Chapter 14
Sections 8 to 12

• Resonance,
• Standing Waves,
• and Musical Instruments

2
Homework Assignment 33

• Problems
• Chapter 14, Problem 34 on page 493
• Chapter 14, Problem 40 on page 493
• Chapter 14, Problem 44 on page 494
• Chapter 14, Problem 49 on page 494

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Natural Frequency of Vibration

• Systems have inertial and elastic characteristics
  which make them want to oscillate or vibrate at
  certain frequencies.
• This specific frequencies are called natural
  frequencies.
• Example A mass on a spring.

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Forced Vibrations

• It's possible to make systems vibrate at other
  frequencies than their natural frequency by
  subjecting them to a forcing vibration of
  frequency, ?f.

FA sin(?ft)
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Forced Vibrations

• At low forcing frequencies, the body vibrate in
  phase with the oscillating force.
• At high forcing frequencies, the body vibrates
  out of phase with the oscillating force.

x
F
x
F
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Multiple Natural Frequencies

• A spring-mass system model has a single natural
  frequency.
• However, real systems often have many natural
  frequencies.
• Today we will look at several different systems
  which have multiple natural frequencies.
• Strings
• Columns of air
• Vibrating Plates
• Guitar body

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Traveling Wave in a string

• A wave traveling through a medium in one
  direction.
• It has frequency, wavelength, and velocity.

v
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Standing Wave Demo

• Starting with low frequency, gradually start
  oscillating the string at faster and faster
  frequencies.

Website http//www.phy.hk/wiki/englishhtm/StatWav
e.htm
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Standing Waves

• When a traveling wave reflects back on itself, it
  creates traveling waves in both directions
• The wave and its reflection interfere according
  to the superposition principle
• With exactly the right frequency, the wave will
  appear to stand still
• This is called a standing wave

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Standing Waves, cont

• A node occurs where the two traveling waves have
  the same magnitude of displacement, but the
  displacements are in opposite directions
• Net displacement is zero at that point
• The distance between two nodes is ½?
• Nodes also occur at each end of the string.
• An antinode occurs where the standing wave
  vibrates at maximum amplitude.

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Standing Waves, cont.

• The pink arrows indicate the direction of motion
  of the parts of the string
• All points on the string oscillate together
  vertically with the same frequency, but different
  points have different amplitudes of motion

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Mathematics of StandingWaves on String

• Fundamental Frequency the lowest natural
  frequency of vibration
• 2nd natural frequency
• 3rd natural frequency

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Mathematics of StandingWaves on String

• Notice that there is a pattern to how the natural
  frequencies increase.

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Mathematics of StandingWaves on String

• This gradual increase in frequency for each new
  natural frequency can be expressed using a single
  equation where

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Standing Waves on a String Frequencies

• 1, 2, 3 form a harmonic series
• 1 is the fundamental and also the first
  harmonic
• 2 is the second harmonic
• Waves in the string that are not in the harmonic
  series are quickly damped out
• In effect, when the string is disturbed, it
  selects the standing wave frequencies

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Harmonics vs. Overtones

• A harmonic is an overtone which is an integral
  multiple of the fundamental frequency.
• A overtone is any natural frequency above the
  fundamental, it may not be a harmonic.
• Not all instruments will have all the harmonics.
• For stringed instruments
• 1st Harmonic Fundamental frequency
• 2nd Harmonic 1st Overtone
• 3rd Harmonic 2nd Overtone
• etc

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String vibration demo

• Harmonics of a String demo at
• http//www.falstad.com/loadedstring/

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An Example of Resonance

• Pendulum A is set in motion
• The others begin to vibrate due to the
  vibrations in the flexible beam
• Pendulum C oscillates at the greatest amplitude
  since its length, and therefore frequency,
  matches that of A

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Other Examples of Resonance and Standing Waves

• Shattering glasses by force vibrationhttp//www.
  youtube.com/watch?v17tqXgvCN0E
• Modes of a Vibrating Square Platehttp//www.youtu
  becom/watch?vZkox6niJ1WcNR1
• Tacoma Narrows Bridge collapse due to
  oscillations by the windhttp//en.wikipedia.org/w
  iki/Tacoma_Narrows_Bridge

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Standing Waves in Air Columns

• Musical Wind Instruments are other devices which
  work by creating standing waves and amplifying
  natural frequencies of a body.
• Brass, Reed, and Woodwinds

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Two common types of Standing Waves in Air Column
Examples

• Open-Open end device
• If the end is open, the elements of the air have
  complete freedom of movement and an antinode
  exists
• Open-Closed end device
• If one end of the air column is closed, a node
  must exist at this end since the movement of the
  air is restricted

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Tube Open at Both Ends
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Resonance in Air Column Open at Both Ends

• In a pipe open at both ends, the natural
  frequency of vibration forms a series whose
  harmonics are equal to integral multiples of the
  fundamental frequency.

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Tube Closed at One End
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Resonance in an Air Column Closed at One End

• The closed end must be a node. The open end is an
  antinode.
• There are no even multiples of the fundamental
  harmonic

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Overtones vs. Harmonics for Open Column
Instruments

• Open-Closed End
• Fundamental 1st Harmonic
• 1st Overtone 3rd Harmonic
• 2nd Overtone 5th Harmonic
• 3rd Overtone 7th Harmonic

• Open-Open End
• Fundamental 1st Harmonic
• 1st Overtone 2nd Harmonic
• 2nd Overtone 3rd Harmonic
• 3rd Overtone 4th Harmonic

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Beats

• Beats are alternations in loudness, due to
  interference of sound waves.
• Waves have slightly different frequencies and the
  time between constructive and destructive
  interference alternates.
• The beat frequency equals the difference in
  frequency between the two sources

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Beats Demo

• Play mysound1.m file
• Equation for beats is the addition of two waves
  with closely related frequencies

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Characteristics of Musical Sound

• Pitch The highness or lowness of a sound.
  Depends upon the frequency of the sound wave
• Intensity (or volume) the relative loudness or
  softness of a sound. Depends upon the amplitude
  of the sound wave
• Timbre the relative richness of a sound.
  Depends upon the mixture of the harmonics or
  overtones which are present in the sound wave.

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Quality of Sound Tuning Fork

• Tuning fork produces only the fundamental
  frequency

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Quality of Sound Flute

• The same note played on a flute sounds
  differently
• The second harmonic is very strong
• The fourth harmonic is close in strength to the
  first

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Quality of Sound Clarinet

• The fifth harmonic is very strong
• The first and fourth harmonics are very similar,
  with the third being close to them

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Example 1

• A stretched string of 40 gram, 8 m long is
  stretched with a tension of 49 N and fixed at
  both ends.
• a) Determine position of nodes and antinodes of
  the 3rd harmonic.
• b) What is the vibration frequency of this
  harmonic.

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Example 1

• m0.040 kg L 8 m F 49 N

3rd Harmonic n 3
8 m
2.67 m
2.67 m
2.67 m
1.33 m
2.67 m
2.67 m
1.33 m
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Example 1
3rd Harmonic n 3
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Example 2

• The human ear canal is about 2.8 cm long. If it
  is regarded as a tube that is open at one end and
  closed at the eardrum, what is the fundamental
  frequency around which we would expect hearing to
  be most sensitive?
• Use speed of sound 340 m/s

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Example 2

• L 2.8 cm 0.028 m v 340 m/s
• Use
• for fundamental n 1

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Example 3

• a) What will be the beat frequency if middle C
  and C are played together?
• b) Will this be audible and distinguishable?
• c) Answer the same questions if the two notes
  are played two octaves below.

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Example 3

• a) What will be the beat frequency if middle C
  and C are played together?
• c) At 2 octaves below? An octave cut the
  frequency by ½, so two octaves would cut the
  frequency by ¼.

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Example 3

• a) Example of playing these sounds
• Matlab file mysound5A.m
• Matlab file mysound5B.m