Dr' James van Howe

1
PH 105
Resonance, Interference, Superposition
Dr. James van Howe Lecture 7
2
Right now we have a test scheduled for Monday.
Shall we delay it till Wednesday?

• Yes, delay it.
• No, bring it on.

3
Name the Mozart Piece

• Jupiter
• Impresario
• Eine kleine Nachtmusik
• Sinfonia Concertante

4
Resonance in Tubes
Which picture accurately shows the standing wave
for the fundamental mode in a open-closed tube?
A
C
B
D
5
Resonance in Tubes
Which picture accurately shows the standing waves
for the fundamental mode in a Open-Open tube?
A
C
B
D
6
True or False
Most of the sound generated by a violin or guitar
comes from vibration of the strings
7
These three sine waves are the same frequency and
have the same amplitude A. What is the new
amplitude when I add them together?
A

• A
• 2A
• 0
• Not that easy

8
These three sine waves are the same frequency and
have the same amplitude A. What is the new
amplitude when I add them together?
A

• A
• 2A
• 0
• Not that easy

9
Beats
the beat frequency is
If and

• 2.5 Hz
• 5 Hz
• 442.5 Hz
• 885 Hz

10
Finish Activity
Demo Maybe String again,
11
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
12
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

13
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
14
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

15
Modes on a string Review
Fundamental Mode
Antinode
Node
Node
l/2
Second Mode
A
A
N
N
N
l
16
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
18
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

19
Resonant Modes of Drum Membrane
Demo Chladni patterns
Fundamental
second mode
third mode
fourth mode
http//mat140.bham.ac.uk/richard/talks/bessel/dru
m_harmonics/table.html
20
Vibrating Bar free ends
fundamental
A
N
N
21
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)

Demo Whirly tube
22
Open Tube
Fundamental
L
l/2
l
3l/2
In general
similar to the string next mode next harmonic
23
Closed Tube
L
l/4
3l/4
5l/4
In general
only odd harmonics exist
24
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
25
Interference The property of waves to add up
(superpose) constructively or destructively
DP
Pressure, DP
position, x
Pressure Change
Video Demo Waves on a coil spring
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Constructive Interference
Two speakers facing each other play the same
note, in phase, at the same amplitude
DP
position, x
Pressure, DP
Pressure Change
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Destructive Interference
Two speakers facing each other play the same
note, at the same amplitude, but 180 degrees out
of phase
DP
position, x
Pressure, DP
Pressure Change
28
Examples of intentional acoustical destructive
interference

• Muffler in car/resonator
• Pipe length chosen so that reflections interfere
  with noise 180 degrees out of phase
• Noise canceling headphones
• Records noise and plays it back 180 degrees out
  of phase for cancellation

Demo video destructive interference