Music 170: Waves
Tamara Smyth, [email protected]
Department of Music,
University of California, San Diego (UCSD)
October 18, 2016
1
Waves
• Recall, a wave is a disturbance or variation that
transfers energy progressively from one point to
another.
– the medium through which the wave travels
experiences local oscillations;
– the particles in the medium do not travel with the
wave.
• Pulse waves traveling on a string: as the wave
travels from left to right,
– the string is displaced up and down;
– the string does not experience any net motion.
– See animation: wave pulse traveling on a string
Music 170: Waves
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Superposition of Waves
• When two (or more) waves travel through the same
medium at the same time,
– they pass through each other without being
disturbed;
– See animation: superposition of waves
• The principle of superposition: the displacement
of the medium at any point in space or time is the
sum of the individual wave displacements.
– holds true for waves that are finite in length (wave
pulses shown here) or continuous (e.g. sine waves);
– the medium must be nondispersive (all
frequencies travel at the same speed).
Music 170: Waves
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Constructive and Destructive
Interference
• The sum of waves in the same medium leads to
interference.
• The nature of the interference depends on the phase
of the waves being added:
– constructive interference: the two waves are in
phase and their sum leads to an increase of
amplitude
– destructive interference: the two waves are out
of phase and the sum leads to a decrease in the
overall amplitude.
– See animation: interference
Music 170: Waves
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Adding Sinusoids of the Same
Frequency
• Adding sinusoids of the same frequency (even with
different amplitudes and phases) produces a sinusoid
of the same frequency.
Sum (red) of Two 4−Hz Sinusoids (blue, green)
Amplitude
15
10
5
0
−5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time (s)
y−axis
• This may also be viewed as vector addition:
V
U
U
V
x−axis
• Vectors of the same frequency, and their sum, rotate
as a unit and thus have the same frequency.
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Reflections at Boundaries
• So how do we get two waves in the same medium?
• When a wave is confined to a medium, such as on a
string, it will reflect upon reaching a boundary.
incident wave
Boundary
reflection
• The wave representing the physical displacement of
the string is the sum of right and left going traveling
waves.
• Wave reflection and standing waves on a rope
Music 170: Waves
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D’Alembert’s Traveling Wave Solution
• This can be seen in D’Alembert’s traveling wave
solution to the wave equation:
y = f1(ct − x) + f2(ct + x).
where
f1(ct − x) , a wave traveling to the right
f2(ct + x) , a wave traveling to the left
with velocity c.
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Reflection form a fixed end
• D’Alembert’s traveling wave solution:
y = f1(ct − x) + f2(ct + x).
• At a fixed end:
– if the string is fixed at x = 0, then y = 0 and
f1(ct) + f2(ct) = 0
f1(ct) = −f2(ct).
– thus the displacement wave is inverted upon
reflection.
– See animation: reflection from a fixed boundary
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Reflection from a free end
• D’Alembert’s traveling wave solution:
y = f1(ct − x) + f2(ct + x).
• At a free end:
– if the string is free at x = 0, then the slope of y
at that position is zero (∂y/∂x = 0) and
−f1(ct) + f2(ct) = 0
f1(ct) = f2(ct).
– thus the displacement wave is not inverted upon
reflection.
– See animation: reflection from a free boundary
Music 170: Waves
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Standing Waves
• Reflected traveling waves cause constructive and
destructive interference leading to standing waves:
– See animation: standing wave
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Pattern of Nodes and Antinodes
• A standing wave is a pattern of alternating nodes and
antinodes (notice no adjacent nodes or antinodes).
– See animation: pattern of nodes and antinodes
• The fundamental mode of oscillation, is determined
by the shortest node-antinode pattern.
• See animation:
standing waves created from a fixed boundary
• See animation:
standing waves created from a free boundary
Music 170: Waves
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String Fixed at Both Ends
• Fixing a string at both ends forces a node at these
positions.
– thus, the shortest standing wave pattern is
node-antinode-node.
– This fundamental mode of oscillation corresponds
to the fundamental frequency or first harmonic.
• The first 3 standing waves (harmonics):
• For string length L and each harmonic number n,
2L
,
– the wavelength is λn =
n
c
cn
– the frequency is fn = =
= nf1.
λ 2L
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Open Tube
• Displacement waves in a tube open at both ends:
Figure 1: Standing displacement waves in an open tube.
• Though opposite in phase to displacement on a
string, the harmonics follow the same pattern for tube
length L:
2L
– the wavelength is λn =
.
n
c
cn
– the fundamental frequency is fn = =
= nf1.
λ 2L
• Pressure waves are opposite phase to displacement
waves.
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Closed Tube
• Displacement waves in tube closed at one end:
Figure 2: Standing waves for pressure in an closed tube
• The wavelength of the first harmonic is λ1 = 4L.
• The next shortest standing wave (creating the next
possible harmonic) is created by adding an antinode
and node following the first node producing:
– pattern: node-antinode-node-antinode,
4L
,
– a harmonic having a wavelength of
3
– the “third” harmonic (with the “second” missing!).
Music 170: Waves
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Odd harmonics in a closed tube
• The next shortest standing wave harmonic has a
4L
.
wavelength of
5
Figure 3: Standing waves for pressure in an closed tube
• Thus, the wavelength can be generalized as
4L
λ=
,
n
and the frequency
cn
fn =
,
4L
for odd harmonics (n = 1, 3, 5, ...).
Music 170: Waves
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