Transcript Physics 207: Lecture 2 Notes
Lecture 29
• • Goals: Chapter 20 Work with a few important characteristics of sound waves. (e.g., Doppler effect) Chapter 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.
• Assignment HW12, Due Friday, May 8 th Thursday, Finish up, begin review for final, evaluations Physics 207: Lecture 29, Pg 1
Doppler effect, moving sources/receivers
Physics 207: Lecture 29, Pg 2
Doppler effect, moving sources/receivers
If the source of sound is moving Toward the observer seems smaller
f
observer Away from observer seems larger
f
observer 1
f
source
v s v
1
f
source
v s v
If the observer is moving Toward the source seems smaller
f
observer 1 v o v
f
source Away from source seems larger
f
observer 1 v o v
f
source Doppler Example Audio Doppler Example Visual Physics 207: Lecture 29, Pg 3
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 = 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.)
t 0
30 m
A
Physics 207: Lecture 29, Pg 4
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: f wall = 340 / (1-20/340) = 361 Hz
f
observer 1
f
source
v s v
Wall becomes “source” for the subsequent part At the speaker f ’ = f wall (1+ 20/340) = 382 Hz
f
observer 1 v o v
f
source
t 0 t 1
30 m Physics 207: Lecture 29, Pg 5
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.) v cart D t + v sound D t = 2 x 30 m = 60 m D t = 60 / (340+20) = 0.17 s d sound = 340 * 0.17 m = 58 m I = 1 / (4 p 58 2 ) = 2.4 x 10 -5 W/m 2 or 74 dBs
t 0 t 1
30 m Physics 207: Lecture 29, Pg 6
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 7
Superposition
Q:
What happens when two waves “collide” ?
A:
They ADD together!
We say the waves are “superimposed”.
Physics 207: Lecture 29, Pg 8
Interference of Waves
2D Surface Waves on Water In phase sources separated by a distance
d d
Physics 207: Lecture 29, Pg 9
Principle of superposition
The superposition of 2 or more waves is called
interference
Destructive interference: Constructive interference: These two waves are in phase .
These two waves are out of phase.
Their crests are aligned.
The crests of one are aligned with the troughs of the other.
Their superposition produces a wave with amplitude 2a Their superposition produces a wave with zero amplitude Physics 207: Lecture 29, Pg 10
Interference: space and time
Is this a point of constructive or destructive interference?
What do we need to do to make the sound from these two speakers interfere constructively?
Physics 207: Lecture 29, Pg 11
Interference of Sound
Sound waves interfere, just like transverse waves do. The resulting wave (displacement, pressure) is the sum of the two (or more) waves you started with.
D
r
|
r
1 | |
r
2 |
D
(
r
2 ,
t
)
D
(
r
1 ,
t
) Maximum constructi ve interferen ce
A
cos[ 2 p (
r
2
r
2
A
cos[ 2 p (
r
1
r
1 / /
t
t
/
T
) 2 ] /
T
) 1 ] D 2 p D 2 p D
r
1 D
r
2 p 2 ( 1 2 p
m
2 )
m
Maximum destructiv e interferen ce D 2 p D
r
m
0 , 1 , 2 ,...
1 2 2 p (
m
1 2 ) D
r
Physics 207: Lecture 29, Pg 12
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 = 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)?
t 0 t 1 d B t 0 C D h
The distances AD and BCD have equal transit times so the sound waves will be in phase. The only need is for AB = Physics 207: Lecture 29, Pg 13
Example
Interference
The geometry dictates everything else.
AB = AC = AD = BC+CD AB+BC = BC + (h 2 = +BC = (h 2 + (d/2) + d/2 2 ) ½ 2 ) ½ = d Eliminating BC gives + 2 +d = 2 (h 2 d + d 2 = 4 h 2 + d + d 2 1 + 2d = 4 h 2 / 2 /4) ½ d = 2 h 2 / – ½
t 0
= 7.5 m
t 1 B t 0
3.25
7.5
C
4.25
D
Because the ground is more dense than air there will be a phase change of p and so we really should set AB to /2 or 0.5 m.
Physics 207: Lecture 29, Pg 14
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 following best represents the resultant wave at this moment?
Original wave (the other has a different phase) (A) (B) (C) (D) (E) Physics 207: Lecture 29, Pg 15
Wave motion at interfaces Reflection 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 Physics 207: Lecture 29, Pg 16
Animation
Reflection of a Wave, Fixed End
Physics 207: Lecture 29, Pg 17
Reflection of a Wave, Free End
Animation
Physics 207: Lecture 29, Pg 18
Standing waves
Two waves traveling in opposite direction interfere with each other.
If the conditions are right,
same k & w , their interference generates a standing wave: D Right (x,t)= a sin(kx w t) D Left (x,t)= a sin(kx+ w t) A standing wave does not propagate in space, it “stands” in place.
A standing wave has nodes and antinodes
Anti-nodes
D(x,t)= D L (x,t) + D R (x,t) D(x,t)= 2a
sin(kx)
cos( w t) The outer curve is the amplitude function
A(x) = 2
a
sin(kx)
when w t = 2 p n n = 0,1,2,…
k
= wave number =
2
π/λ
Nodes
Physics 207: Lecture 29, Pg 21
Standing waves on a string
Longest wavelength allowed is one half of a wave Fundamental: /2 = L = 2 L
m
2
L m
m
1 , 2 , 3 ,...
f v m
Recall
v = f
f m
m v
2
L
Overtones
m
> 1
Physics 207: Lecture 29, Pg 22
Vibrating Strings- Superposition Principle
Violin, viola, cello, string bass Guitars Ukuleles Mandolins Banjos
D(x,0)
Physics 207: Lecture 29, Pg 23
Standing waves in a pipe
Open 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: /2 /2 /4 Physics 207: Lecture 29, Pg 24
m
2
L m f m m
m v
2
L
1 , 2 , 3 ,...
Standing waves in a pipe
m
2
L m
m
4
L m f m m
m v
2
L
1 , 2 , 3 ,...
f m m
m v
4
L
1 , 3 , 5 ,...
Physics 207: Lecture 29, Pg 25
Combining Waves
Fourier Synthesis
Physics 207: Lecture 29, Pg 26
Lecture 29
• Assignment HW12, Due Friday, May 8 th Physics 207: Lecture 29, Pg 27