HW12, Due Friday, May 8 th Thursday, Finish up, begin ... Due Friday, May 8 th Thursday, Finish up,...
Transcript of HW12, Due Friday, May 8 th Thursday, Finish up, begin ... Due Friday, May 8 th Thursday, Finish up,...
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Physics 207 – Lecture 29
Physics 207: Lecture 29, Pg 1
Lecture 29Goals:Goals:
•• Chapter 20Chapter 20� Work with a few important characteristics of sound waves.
(e.g., Doppler effect)
•• Chapter 21Chapter 21� Recognize standing waves are the superposition of two
traveling waves of same frequency� Study the basic properties of standing waves� Model interference occurs in one and two dimensions� Understand beats as the superposition of two waves of
unequal frequency.
•• AssignmentAssignment� HW12, Due Friday, May 8th
� Thursday, Finish up, begin review for final, evaluations
Physics 207: Lecture 29, Pg 2
Doppler effect, moving sources/receivers
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Physics 207 – Lecture 29
Physics 207: Lecture 29, Pg 3
Doppler effect, moving sources/receivers
� If the source of sound is moving� Toward the observer ⇒
λ seems smaller
� Away from observer ⇒λ seems larger
� If the observer is moving� Toward the source ⇒
λ seems smaller
� Away from source ⇒λ seems larger
vvs
ff
−=
1source
observer
sourceo
observer vv
1 ff
+=
Doppler Example Audio
Doppler Example Visual
vvs
ff
+=
1source
observer
sourceo
observer vv
1 ff
−=
Physics 207: Lecture 29, Pg 4
Doppler Example � A speaker sits on a small moving cart and emits a short 1
Watt sine wave pulse at 340 Hz (the speed of sound in air is 340 m/s, so λ = 1m ). The cart is 30 meters away from the wall and moving towards it at 20 m/s.
� The sound reflects perfectly from the wall. To an observer on the cart, what is the Doppler shifted frequency of the directly reflected sound?
� Considering only the position of the cart, what is the intensity of the reflected sound? (In principle on would have to look at the energy per unit time in the moving frame.)
t0
A
30 m
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Physics 207 – Lecture 29
Physics 207: Lecture 29, Pg 5
Doppler Example� The sound reflects perfectly from the wall. To an observer on
the cart, what is the Doppler shifted frequency of the directly reflected sound?
At the wall: fwall = 340 / (1-20/340) = 361 Hz
Wall becomes “source” for the subsequent part
At the speaker f ’ = fwall (1+ 20/340) = 382 Hz
t030 m
t1
vvs
ff
−=
1source
observer
sourceo
observer vv
1 ff
+=
Physics 207: Lecture 29, Pg 6
Example Interference
� Considering only the position of the cart, what is the intensity of the reflected sound to this observer? (In principle one would have to look at the energy per unit time in the moving frame.)
vcart ∆t + vsound ∆t = 2 x 30 m = 60 m
∆t = 60 / (340+20) = 0.17 s � dsound = 340 * 0.17 m = 58 m
I = 1 / (4π 582) = 2.4 x 10-5 W/m2 or 74 dBs
t030 m
t1
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Physics 207 – Lecture 29
Physics 207: Lecture 29, Pg 7
Doppler effect, moving sources/receivers
� Three key pieces of information
� Time of echo
� Intensity of echo
� Frequency of echo
Plus prior knowledge of object being studied
� With modern technology (analog and digital) this can be done in real time.
Physics 207: Lecture 29, Pg 8
Superposition
� Q: What happens when two waves “collide” ?
� A: They ADD together!� We say the waves are “superimposed”.
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Physics 207 – Lecture 29
Physics 207: Lecture 29, Pg 9
Interference of Waves� 2D Surface Waves on Water
In phase sources separated by a distance d
d
Physics 207: Lecture 29, Pg 10
Principle of superposition
� The superposition of 2 or more waves is called interference
Constructive interference:These two waves are in phase.Their crests are aligned.
Their superposition produces a
wave with amplitude 2a
Destructive interference:These two waves are out of
phase.The crests of one are aligned
with the troughs of the other.
Their superposition produces a
wave with zero amplitude
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Physics 207 – Lecture 29
Physics 207: Lecture 29, Pg 11
Interference: space and time
� Is this a point of constructiveor destructive interference?
What do we need to do to make the sound from these two speakers interfere constructively?
Physics 207: Lecture 29, Pg 12
Interference of SoundSound waves interfere, just like transverse waves do. The resulting wave (displacement, pressure) is the sum of the two (or more) waves you started with.
|||| 21 rrrrr −=∆
,...2,1,0
)21( 22
ceinterferen edestructiv Maximum
)(22
22
ceinterferen veconstructi Maximum
21
21
21
=
+=−+∆=∆
=−+∆=∆
=−+∆=∆
m
mr
mr
mr
πφφλπφ
λφφπλφ
πλ
πφφλπφ
])//(2cos[),( 22
2
2 φλπ +−= Ttrr
AtrD
])//(2cos[),( 11
1
1 φλπ +−= Ttrr
AtrD
r∆
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Physics 207 – Lecture 29
Physics 207: Lecture 29, Pg 13
Example Interference� A speaker sits on a pedestal 2 m tall and emits a sine wave
at 343 Hz (the speed of sound in air is 343 m/s, so λ = 1m ). Only the direct sound wave and that which reflects off the ground at a position half-way between the speaker and the person (also 2 m tall) makes it to the persons ear.
� How close to the speaker can the person stand (A to D) so they hear a maximum sound intensity assuming there is no phase change at the ground (this is a bad assumption)?
The distances AD and BCD have equal transit times so the sound waves will be in phase. The only need is for AB = λ
t1
t0
t0
AB
A DC
d
h
Physics 207: Lecture 29, Pg 14
Example Interference
� The geometry dictates everything else.
AB = λ AD = BC+CD = BC + (h2 + (d/2)2)½ = dAC = AB+BC = λ +BC = (h2 + d/22)½
Eliminating BC gives λ+d = 2 (h2 + d2/4)½
λ + 2λd + d2 = 4 h2 + d2
1 + 2d = 4 h2 / λ � d = 2 h2 / λ – ½= 7.5 m
t1
t0
t0
AB
A DC
7.5
4.253.25
Because the ground is more dense than air there will be a phase change of π and so we really should set AB to λ/2 or 0.5 m.
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Physics 207 – Lecture 29
Physics 207: Lecture 29, Pg 15
Exercise Superposition
� Two continuous harmonic waves with the same frequency and amplitude but, at a certain time, have a phase difference of 170°are superimposed. Which of the follo wing best represents the resultant wave at this moment?
(A)
(E)
(D)
(C)
(B)
Original wave (the other has a different phase)
Physics 207: Lecture 29, Pg 16
Wave motion at interfacesReflection of a Wave, Fixed End
� When the pulse reaches the support, the pulse moves back along the string in the opposite direction
� This is the reflection of the pulse
� The pulse is inverted
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Physics 207 – Lecture 29
Physics 207: Lecture 29, Pg 17
Reflection of a Wave, Fixed EndAnimation
Physics 207: Lecture 29, Pg 18
Reflection of a Wave, Free End
Animation
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Physics 207 – Lecture 29
Physics 207: Lecture 29, Pg 19
Transmission of a Wave, Case 1
� When the boundary is intermediate between the last two extremes ( The right hand rope is massive or massless.) then part of the energy in the incident pulse is reflected and part is transmitted
� Some energy passes through the boundary
� Here µrhs > µlhs
Animation
Physics 207: Lecture 29, Pg 20
Transmission of a Wave, Case 2
� Now assume a heavier string is attached to a light string
� Part of the pulse is reflected and part is transmitted� The reflected part is not inverted
Animation
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Physics 207 – Lecture 29
Physics 207: Lecture 29, Pg 21
Standing waves� Two waves traveling in opposite direction interfere with each
other.
If the conditions are right, same k & ω, their interference generates a standing wave:
DRight(x,t)= a sin(kx-ωt) DLeft(x,t)= a sin(kx+ωt)A standing wave does not propagate in space, it “stands” in place.A standing wave has nodes and antinodes
D(x,t)= DL(x,t) + DR(x,t)
D(x,t)= 2a sin(kx) cos(ωt)The outer curve is the
amplitude function A(x) = 2a sin(kx)
when ωt = 2πn n = 0,1,2,…k = wave number = 2�/
�Nodes
Anti-nodes
Physics 207: Lecture 29, Pg 22
Standing waves on a string
� Longest wavelength allowed is one half of a wave
Fundamental: λ/2 = L � λ = 2 L
,...3,2,1
2
=
==
m
fv
mL
mmλ
Recall v = f λ
Lvmfm 2
=Overtones m > 1
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Physics 207 – Lecture 29
Physics 207: Lecture 29, Pg 23
Vibrating Strings- Superposition Principle
� Violin, viola, cello, string bass� Guitars� Ukuleles� Mandolins� Banjos
D(x,0)
An
tin
od
eD
(0,t
)
Physics 207: Lecture 29, Pg 24
Standing waves in a pipeOpen end: Must be a displacement antinode (pressure minimum) Closed end: Must be a displacement node (pressure maximum) Blue curves are displacement oscillations. Red curves, pressure.
Fundamental: λ/2 λ/2 λ/4
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Physics 207 – Lecture 29
Physics 207: Lecture 29, Pg 25
Standing waves in a pipe
,...3,2,1
2
2
=
=
=
m
Lvmf
mL
m
mλ
,...3,2,1
2
2
=
=
=
m
Lvmf
mL
m
mλ
,...5,3,1
4
4
=
=
=
m
Lvmf
mL
m
mλ
Physics 207: Lecture 29, Pg 26
Combining Waves
Fourier Synthesis