the two displacements at a particular location vs. timewoolf/2020_jui/mar25.pdfe.g. a medieval...
TRANSCRIPT
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17.4 Beats
Two overlapping waves with slightly different frequencies gives rise to the phenomena of beats.
1 Traveling waves with slightly different frequencies
The two displacements at a particular location vs. time
http://www.kettering.edu/physics/drussell/Demos/superposition/super4.gif
http://www.kettering.edu/physics/drussell/Demos/superposition/super4.gif
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17.4 Beats
The beat frequency is the difference between the two sound frequencies. 2
The math of beats At a particular location:
)2sin()2sin()()()( 2121 tfAtfAtytyty ππ +=+=
We now make use of a trigonometric identity:
[ ]
( )tftfAy
tfftffA
tfAtfAty
2sin2
2cos2
22sin
22cos2
)2sin()2sin()(
2121
21
ππ
ππ
ππ
∆=⇒
+
−=
+=An identity is an equation that is always true regardless of the value(s) of the arguments involved.
Result is a wave with a frequency equal to the average of f1 and f2, but modulated by a sinusoidal envelope of frequency (f1- f2)/2. But the sound is loudest when the envelope function is either maximum or minimum.
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Multi-Concept Example: Two speakers are set up along the x-axis 25 m apart, both playing a 512 Hz pure tone. Batman is walking at speed u away from one speaker and toward the second. At half way between the speakers, he hears a beat frequency between the two speakers at 4.0 Hz. The speed of sound is 343 m/s. At what speed u is Batman walking?
17.4 Beats
1 2 u
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4
Multi-Concept Example: Two speakers are set up along the x-axis 25 m apart, both playing a 512 Hz pure tone. Batman is walking at speed u away from one speaker and toward the second. At half way between the speakers, he hears a beat frequency between the two speakers at 4.0 Hz. The speed of sound is 343 m/s. At what speed u is Batman walking?
17.4 Beats
Solution: Batman represents a moving observer away from stationary source 1, so he hears frequency f1 from speaker 1:
−=
vuff s 11
And: Batman represents a moving observer toward stationary source 2, so he hears frequency f2 from speaker 2:
+=
vuff s 12
The sound from the two speakers will interfere, and produce beats of frequency
Hz)2(512 Hz)m/s)(4.0 343(
22
2
1121
==→=→
−=−−−=
+−
−=−=∆=
s
BsB
sssss
ssB
fvfuf
vuf
fvuf
vuff
vuf
vuf
vufffff
m/s 34.1=⇒ u
1 2 u
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5
17.5 Transverse Standing Waves
Two waves with the same amplitude and frequency traveling in opposite directions on the SAME medium interfere in such a way as to produce a stationary oscillating pattern This is called a Standing Wave (e.g. the line segment between the two bobs in the ripple tank)
yAgain: the locations where you always have destructive interference are called “nodes”. The locations with maximum constructive interference are “anti-nodes”
[ ])/22sin()/22sin(),(),(),()/22sin(),( ),/22sin(),(
λππλππλππλππ
xftxftAtxytxytxyxftAtxyxftAtxy++−=+=
+=−=
−+
−+
The math of Standing Waves: the rightward and leftward waves are given, respectively:
We again use the identity:
[ ] )2sin()/2cos(2),()2sin()/2cos(2),(ftxAtxy
ftxAtxyπλπ
πλπ=⇒
−=
http://www.kettering.edu/physics/drussell/Demos/superposition/super3.gif
http://www.kettering.edu/physics/drussell/Demos/superposition/super3.gif
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17.5 Transverse Standing Waves
It is possible to produce self-sustaining standing waves on nearly ideal (very little energy loss as the wave propagates) by taking advantage of reflections There are two special cases of reflections: (1) Reflection from a fixed end (where y=0 always)
Click to start movie http://www.youtube.com/watch?v=LTWHxZ6Jvjs
http://www.kettering.edu/physics/drussell/Demos/reflect/hard.gif
http://www.youtube.com/watch?v=LTWHxZ6Jvjshttp://www.kettering.edu/physics/drussell/Demos/reflect/hard.gif
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17.5 Transverse Standing Waves
(2) Reflection from a free end (where the slope of the tangent ∆y/∆x=0 always)
Click to start movie
http://www.youtube.com/watch?v=aVCqq5AkePI
http://www.kettering.edu/physics/drussell/Demos/reflect/soft.gif
http://www.youtube.com/watch?v=aVCqq5AkePIhttp://www.kettering.edu/physics/drussell/Demos/reflect/soft.gif
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17.5 Transverse Standing Waves
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By providing small impulses at one end, one can generate a large standing wave from the reflections (the hand in this case is very nearly a node), provided that the frequency is right , such that the end (boundary) conditions are satisfied The boundary conditions depend on the type of ends: (1) Fixed end = node (y=0 always) (2) Free end = anti-node (and ∆y/∆x=0 always) Case A: two fixed ends (all string musical instruments are made this way)
When you get the right frequencies where small stimuli maintains a large standing wave: we have what is called resonance All musical instruments depends on resonance to generate sound og a definite pitch/frequency.
http://www.youtube.com/watch?v=jovIXzvFOXo
http://www.youtube.com/watch?v=jovIXzvFOXo
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17.5 Transverse Standing Waves
,4,3,2,1 2
=
= n
LvnfnString fixed at both ends 9
f1 f2=2 f1 f3=3 f1
First Harmonic n = 1
Second Harmonic n = 2
Third Harmonic n = 3
==→=
LvvfL
21
2 11
1
λλ
==→=
LvvfL
22
22
12
2
λλ
==→=
LvvfL
23
23
13
3
λλ
L=length of string The progression of harmonics adds ONE node at a time
http://www.physicsclassroom.com/class/waves/u10l4eani1.gif
http://www.physicsclassroom.com/class/waves/u10l4eani1.gif
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17.5 Transverse Standing Waves
,4,3,2,1
2
=
=
nLvnfn
10
Guitar players shorten the “open” string to produce sound of a higher frequency than the open string: as L decreases, f increases
Conceptual Example 5 The Frets on a Guitar Frets allow a the player to produce a complete sequence of musical notes on a single string. Starting with the fret at the top of the neck, each successive fret shows where the player should press to get the next note in the sequence. Musicians call the sequence the chromatic scale, and every thirteenth note in it corresponds to one octave, or a doubling of the sound frequency. The spacing between the frets is greatest at the top of the neck and decreases with each additional fret further on down. Why does the spacing decrease going down the neck?
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17.6 Longitudinal Standing Waves
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Longitudinal standing waves can also be generated using a speaker and a spring This set up gives roughly the equivalent situation of two fixed ends (except the bottom end is not quite a node) 2nd harmonic: ~32 Hz: one node in the middle between ends 4th harmonic: ~64 Hz three nodes between ends 6th harmonic: ~96 Hz five nodes between ends
http://www.youtube.com/watch?v=12pjjPIE2IQ
http://www.youtube.com/watch?v=12pjjPIE2IQ
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17.6 Longitudinal Standing Waves
,4,3,2,1 2
=
= n
LvnfnTube open at both ends 12
Most wind instruments use sound waves resonating longitudinally in columns of air. 2 cases: (1) Two open (free) ends e.g. a flute The movie clip shows the equivalent transverse harmonic
=
=
Lvf
L
2
2
1
1λ
=
=
Lvf
L
22
22
2
2λ
http://www.youtube.com/watch?v=7_GeW73SGnc
http://www.youtube.com/watch?v=7_GeW73SGnc
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17.6 Longitudinal Standing Waves
Example: Playing a Flute
When all the holes are closed on one type of flute, the lowest note it can sound is middle C (261.6 Hz). If the speed of sound is 343 m/s, and the flute is assumed to be a cylinder open at both ends, determine the distance L.
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17.6 Longitudinal Standing Waves
Example: Playing a Flute When all the holes are closed on one type of flute, the lowest note it can sound is middle C (261.6 Hz). If the speed of sound is 343 m/s, and the flute is assumed to be a cylinder open at both ends, determine the distance L.
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,4,3,2,1 2
=
= n
Lvnfn
( )( ) m 656.0Hz 261.62
sm34312
===nf
nvL
Method 2: memorize the appropriate formula (of the three cases), and apply it
Method 1: look at the physical situation: open ends anti-nodes at both ends. Lowest frequency: only one node between the ends (and only one node total)
( )( ) m 656.0 Hz261.62
sm3432
22/
221
===
===→=∴
fvL
fvfvLL λλ
Here n=1 for lowest frequency
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17.6 Longitudinal Standing Waves
,5,3,1 4
=
= n
LvnfnTube open at one end 15
(2) One open (free) end e.g. a medieval trumpet The movie clip shows the equivalent transverse harmonics, starting at the 2nd harmonic (n=3). **N=1 is too difficult
=
=
Lvf
L
4
4
1
1λ
=
=
Lvf
L
43
43
3
3λ
Even n modes do not exist
http://www.youtube.com/watch?v=DWhCdlPM19M
http://www.youtube.com/watch?v=DWhCdlPM19M
17.4 Beats17.4 BeatsSlide Number 3Slide Number 4Slide Number 5Slide Number 6Slide Number 717.5 Transverse Standing Waves17.5 Transverse Standing Waves17.5 Transverse Standing Waves17.6 Longitudinal Standing Waves17.6 Longitudinal Standing Waves17.6 Longitudinal Standing Waves17.6 Longitudinal Standing Waves17.6 Longitudinal Standing Waves