Harmonic Superposition and Standing Waves · 2016. 1. 4. · A: The wave speed on a string is given...
Transcript of Harmonic Superposition and Standing Waves · 2016. 1. 4. · A: The wave speed on a string is given...
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Harmonic Superposition and Standing Waves
11 January 2016
PHYC 1290 Department of Physics and Atmospheric Science
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Superposition of Harmonic Waves
y1 y2
y1 = A sin(kx - ωt) y2 = A sin(kx + ωt)⊖ for wave moving right ⊕ for wave moving left
Let them superpose:
➞ y = y1 + y2 = A sin(kx - ωt) + A sin(kx + ωt)
Consider 2 waves with same amplitude, wavelength and period:
A
-A0
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t = π/ωt = 2π/3ω
t = π/3ωt = 0
Use the trig identity
sin B + sin C = 2 sin�
12(B + C)
⇥cos
�12(B � C)
⇥
with B = kx - ωt and C = kx + ωt .
➞ y = 2A sin(kx)cos(-ωt)
But cos(-θ) = cos(θ).
➞ y = 2A sin(kx)cos(ωt)
It doesn’t move left or right!
It is a “standing wave” of amplitude 2A.
y
xπ/k 2π/k2A
-2A0
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Nodes Anti-nodes
Simulation
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How to make a standing wave
fixed end
drivesinusoidally
When the wave is reflected from the fixed end, it will be inverted but will have the same k, ω and A.
➞ y = ... = 2A cos(kx) sin(ωt) (a standing wave)
➞ y = A sin(kx - ωt) - A sin(kx + ωt)
inverted moving leftmoving right
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DemoStanding wave on a spring
By shaking the spring at different frequencies, we can see different “harmonics” m:
m = 1
m = 2
m = 3
m = 4
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DemoStanding longitudinal wave on an aluminum rod
Running fingers along an aluminum rod (using lots of resin) excites longitudinal modes that we can hear.
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LEnds can move(free, not fixed)
Hold herefor m=1
Hold here for m=2
How Does it Work?
yx
Node
m=1:
x
Node Node
y
m=2:
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Standing wave modes:
L
m = 1
m = 2
m = 3
m = 4
Fixed-fixed ends
�2 =2L2
�3 =2L3
�4 =2L4
Harmonic numbers:
f2 = 2v
2L
f3 = 3v
2L
f4 = 4v
2L
�1 =2L1
Wavelengths:Frequencies
(using v = fλ):
f1 = 1v
2L
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GeneralizationFor standing waves with fixed ends, the wavelength for
harmonic (or mode) number m is
�m =2Lm
The lowest frequency (f1) is called the “fundamental harmonic”:
f1 =v
2Lwhich allows us to write
fm = mf1
and the frequency is
fm = mv
2L
with m = 1, 2, 3, ...
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Standing wave modes
L
m = 1
Free-free ends
Harmonic numbers:
m = 2
m = 3
The generalizations are the same as before:
�m =2Lm
f1 =v
2L andfm = mf1, with m = 1, 2, 3, ...
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Standing wave modes
L
m = 1
m = 3
m = 5
Fixed-free ends.
Harmonic numbers:
The generalizations for fixed-free ends are different:
�m =4Lm
f1 =v
4L andfm = mf1, with m = 1, 3, 5, ...
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Extra Material
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Standing wave modes
L
m = 1 �1 =2L1
Free-free ends
Harmonic numbers:
m = 2 �2 =2L2
f2 = 2v
2L
m = 3 �3 =2L3
f3 = 3v
2L
Wavelengths:Frequencies
(using v = fλ):
f1 = 1v
2L
Conclusion: We can use the same generalizations as before.
�m =2Lm
f1 =v
2L andfm = mf1, with m = 1, 2, 3, ...
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Standing wave modes
L
m = 1
m = 3
m = 5
Fixed-free ends
Harmonic numbers: Wavelengths:
Frequencies (using v = fλ):
�1 =4L1
�3 =4L3
�5 =4L5
f1 = 1v
4L
f3 = 3v
4L
f5 = 5v
4L
Conclusion: The generalizations for fixed-free ends are different.
�m =4Lm
f1 =v
4L andfm = mf1, with m = 1, 3, 5, ...
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Standing wave on a stringHere is a 60 Hz standing wave (m=2) on a string. Part way through a strobe light is used to isolate the string’s motion.
http://youtu.be/rkeIubVQ0N8
http://youtu.be/rkeIubVQ0N8
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ExampleMiddle-C (C4) on a piano has fundamental frequency of 262 Hz.
Q: What are the frequencies of the Harmonics?
A: The string is fixed at both ends, so fm = mf1.
m=1: f1 = 262 Hz (given)
... and so on.
m=2: f2 = 2×f1 = 524 Hzm=3: f3 = 3×f1 = 786 Hz
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Q: If the string is 62.5 cm long, what are the wavelengths?
A: For fixed ends, λm = 2L/m , som=1: λ1 = 2L/1 = 125 cmm=2: λ2 = 2L/2 = 62.5 cmm=3: λ3 = 2L/3 = 41.67 cm
Q: What are the wave speeds on the string?
A: Using, v = fλ,m=1: v1 = f1×λ1 = 32750 cm/sm=2: v2 = f2×λ2 = 32750 cm/s m=3: v3 = f3×λ3 = 32750 cm/s
The speeds are all the same! This is expected because the wave speed v depends only on the medium’s properties.
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Q: What tension does the middle-C string have?
A: The wave speed on a string is given by
v =
�T
µ� T = µv2
We need to first determine the linear density μ=m/L.
➞ μ = m/L = density × πr 2 = 0.00614 kg/m
➞ T = 659 N
v =
�T
µ� T = µv2
Piano wire is made of steel which has density 7820 kg/m3. The middle-C string is 1 mm in diameter.
➞ m = density × volume = density × πr 2L
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659 N is a lot of tension for one string! This is equivalent to having the tension on one string
drawn by a 67 kg (148 lb) mass.
There are normally about 230 strings in a piano (although there are 88 keys, some keys have
multiple strings).
This means there is a combined tension of over 150,000 N (15,000 kg or 30,000 lbs) in a piano!!
For this reason, at the heart of a piano is a cast-iron frame (or harp, pictured left) needed to
sustain the massive tension.
Image souce: http://commons.wikimedia.org/wiki/File:Fluegel-Rahmen.jpg
http://commons.wikimedia.org/wiki/File:Fluegel-Rahmen.jpg
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Here is a view inside a grand piano. You can see the strings drawn across the cast-iron frame.
Image source: http://commons.wikimedia.org/wiki/File:Bosendorfer_185.JPG
http://commons.wikimedia.org/wiki/File:Bosendorfer_185.JPG