PH 105

1
PH 105
Resonance
Dr. James van Howe Lecture 9
2
Name the artist

• Randy Newman
• Ray Charles
• Louis Armstrong
• Herbie Hancock

3
True or False
Most of the sound generated by a violin or guitar
comes from vibration of the strings
4
Resonance in Tubes
How many wavelengths do you see?

• ¼ wavelength
• ½ wavelength
• 1 wavelength
• 1 ½ wavelengths

5
Classic Example of Resonance
Pushing a someone on a swing (pendulum)
Little pushes at the right frequency result in a
huge amplitude of the swing
driving force
Right frequency or resonant frequency is when
the pusher drives the swing at its natural
frequency
Spring constant
driving force
Natural Frequency
6
Resonance

• It is typically easy to get an object to vibrate
  at resonant frequencies, and hard at other
  frequencies
• If driven at many frequencies, an object will
  pick out the resonant frequencies and vibrate at
  those frequencies, the rest are filtered out
• Most objects are not the simple mass-on-a-spring
  which has one natural frequency, but are coupled
  oscillators (many masses and springs). They
  therefore have many resonant frequencies or modes
  of vibration

Video demos glider on track, Tacoma Narrows
bridge collapse
7
Response Curve and Quality Factor
2.5
Sharp Resonance
Dull Resonance
high Q
low Q
2
1.5
Amplitude (arbitrary units)
High damping
1
Low damping
0.5
0
0
20
40
60
80
100
120
140
160
180
200
0
20
40
60
80
100
120
140
160
180
200
Driving frequency (Hz)
Driving frequency (Hz)
Quality Factor
Higher Q, higher energy put into vibration
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Resonance in Musical Instruments

• String Instruments modes on a string
• Percussion string-like modes, i.e. drum
  two-dimensional string
• Brass, woodwinds, human voice modes in open and
  closed tubes
• Helmholtz resonator (sound box) cavity modes

9
Modes on a string Review
Fundamental Mode
Antinode
Node
Node
l/2
Second Mode
A
A
N
N
N
l
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Guitar
Bridge
Nut
Neck
Suppose I excite the fundamental mode
L
l/2
The length of the string between the fixed points
sets the wavelength
The velocity is set by the tension and linear
density of the string
or
So the frequency (pitch) is given by
fundamental resonance
How do I go up in frequency by a factor of two
(an octave)?
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L
l/2
l
3l/2
In general
For the string, next resonance is the next
harmonic
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Definitions

• Harmonic whole number multiples of the
  fundamental frequency
• Overtones multiples of the fundamental that are
  not necessarily whole number
• Partials all overtones
• Octave interval between two tones that is a
  ratio of 21

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Resonant Modes of Drum Membrane
Fundamental
second mode
third mode
fourth mode
http//mat140.bham.ac.uk/richard/talks/bessel/dru
m_harmonics/table.html
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Vibrating Bar free ends
fundamental
A
N
N
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Wind Instruments
Open Tube
A
N
N
A
A

• For tube, we draw antinodes an nodes transverse
  for convenience they are really longitudinal!
• Open tube is like a rope with free ends
  displacement must be max there (antinode),
• Closed tube is like a rope with one fixed end no
  displacement there (node)

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Open Tube
Fundamental
L
l/2
l
3l/2
In general
similar to the string next mode next harmonic
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Closed Tube
L
l/4
3l/4
5l/4
In general
only odd harmonics exist
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End Correction
Brief note every open end of a tube is
acoustically longer
End correction add 0.61r where r is the radius,
of the tube to any open end to get the acoustic
length
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Sympathetic Vibration
Demo acoustic vs. electric guitar, tuning fork

• Violin, piano, or guitar strings make very little
  sound
• By connecting the strings to the top plate
  through the bridge, the vibrations are
  transmitted to the plate and soundboard
• Top plate has a much bigger surface area and move
  a much larger volume of air (can think of the
  plate as many little strips or strings vibrating
  constructively) amplifying the sound

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Sympathetic Vibration Violin

• Strings resonate at certain modes (frequencies)
  that you bow or pluck
• Bridge transfers these to the sounding board
  which vibrates at these frequencies (amplifies
  sound)
• Sounding board also has its own resonant
  frequencies in playing range and selects some
  tones at the expense of others (makes each violin
  unique)
• Top plate also sets the air in the sound box in
  motion so that radiates out of the f-holes
  (helmholtz resonator)

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Homework
For Wednesday Read Handout
For Friday Chapter 4 R11, E3 Read Chapter 7 pg
135-144, skim rest
R review questions E Excercise