Transcript Physics 207: Lecture 2 Notes

Lecture 28, Dec. 8
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, Dec. 12th
 For Wednesday, Review for final, Evaluations
Physics 207: Lecture 28, Pg 1
Doppler effect, moving sources/receivers
Physics 207: Lecture 28, Pg 2
Doppler effect, moving sources/receivers
 If the source of sound is moving
f observer
f source

v
1  vs
 Away from observer  f observer
 seems larger
f source

v
1  vs
 Toward the observer 
 seems smaller
 If the observer is moving
 Toward the source  f observer
 seems smaller
 vo 
 1   f source
v

 Away from source 
 vo 
f observer  1   f source
 seems larger
v


Doppler Example Audio
Doppler Example Visual
Physics 207: Lecture 28, Pg 3
Superposition
 Q: What happens when two waves “collide” ?
 A: They ADD together!
 We say the waves are “superimposed”.
Physics 207: Lecture 28, Pg 4
Interference of Waves
 2D Surface Waves on Water
In phase sources separated
by a distance d
d
Physics 207: Lecture 28, Pg 5
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
Physics 207: Lecture 28, Pg 6
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 28, Pg 7
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.
A
D(r2 , t )  2 cos[ 2 (r2 /   t / T )  2 ]


r2
r | r1 |  | r2 |
A
D(r1 , t )  2 cos[ 2 (r1 /   t / T )  1 ]
r1
Maximum constructi ve interferen ce
  2 r  1  2  2 m



  r 
(1  2 )  m
2
2
Maximum destructiv e interferen ce
  2 r  1  2  2 (m  1 )
2

m  0,1,2,...
r
Physics 207: Lecture 28, Pg 8
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)?
t1
t0
d
A
t0
B
D
h
C
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 28, Pg 9
Example Interference

The geometry dictates everything else.
AB = 
AD = BC+CD = BC + (h2 + (d/2)2)½ = d
AC = AB+BC =  +BC = (h2 + d/22)½
Eliminating BC gives
+d = 2 (h2 + d2/4)½
 + 2d + d2 = 4 h2 + d2
1 + 2d = 4 h2 /   d = 2 h2 /  – ½
= 7.5 m
t1
t0
7.5
A
t0
B
3.25 C
4.25
D
Because the ground is more dense than air there will be a phase
change of  and so we really should set AB to /2 or 0.5 m.
Physics 207: Lecture 28, Pg 10
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)
(D)
(C)
(E)
Physics 207: Lecture 28, Pg 11
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 28, Pg 12
Animation
Reflection of a Wave, Fixed End
Physics 207: Lecture 28, Pg 13
Reflection of a Wave, Free End
Animation
Physics 207: Lecture 28, Pg 14
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 mrhs > mlhs
Animation
Physics 207: Lecture 28, Pg 15
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
Physics 207: Lecture 28, Pg 16
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:
DRight(x,t)= a sin(kx-wt) DLeft(x,t)= a sin(kx+wt)
A standing wave does not propagate in space, it “stands” in place.
A standing wave has nodes and antinodes
Anti-nodes
D(x,t)= DL(x,t) + DR(x,t)
D(x,t)= 2a sin(kx) cos(wt)
The outer curve is the
amplitude function
A(x) = 2a sin(kx)
when wt = 2n n = 0,1,2,…
k = wave number = 2π/λ
Physics 207:Nodes
Lecture 28, Pg 17
Standing waves on a string
 Longest wavelength allowed is
one half of a wave
Fundamental: /2 = L   = 2 L
v
2
L
m 

m
fm
m  1,2,3,...
Recall v = f 
v
fm  m
2L
Overtones m > 1
Physics 207: Lecture 28, Pg 18
Vibrating Strings- Superposition Principle
 Violin, viola, cello, string bass
D(x,0)
 Guitars
 Ukuleles
 Mandolins
Antinode D(0,t)
 Banjos
Physics 207: Lecture 28, Pg 19
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 28, Pg 20
Standing waves in a pipe
m  2 L
m  2 L
m  4 L
fm  m v
2L
m  1,2,3,...
fm  m v
2L
m  1,2,3,...
fm  m v
4L
m  1,3,5,...
m
m
m
Physics 207: Lecture 28, Pg 21
Combining Waves
Fourier Synthesis
Physics 207: Lecture 28, Pg 22
Organ Pipe Example
A 0.9 m organ pipe (open at both ends) is measured to have
it’s first harmonic (i.e., its fundamental) at a frequency of
382 Hz. What is the speed of sound (refers to energy
transfer) in this pipe?
L=0.9 m
f = 382 Hz and f  = v with  = 2 L / m (m = 1)
v = 382 x 2(0.9) m  v = 687 m/s
Physics 207: Lecture 28, Pg 23
Standing Waves
 What happens to the fundamental frequency of a pipe, if
the air (v =300 m/s) is replaced by helium (v = 900 m/s)?
Recall: f  = v
(A) Increases
(B) Same
(C) Decreases
Physics 207: Lecture 28, Pg 24
Superposition & Interference
 Consider two harmonic waves A and B meet at t=0.
 They have same amplitudes and phase, but
Beat Superposition
w2 = 1.15 x w1.
 The displacement versus time for each is shown below:
A(w1t)
B(w2t)
C(t) = A(t) + B(t)
DESTRUCTIVE
INTERFERENCE
CONSTRUCTIVE
INTERFERENCE
Physics 207: Lecture 28, Pg 25
Superposition & Interference
 Consider A + B,
yA(x,t)=A cos(k1x–w1t)
yB(x,t)=A cos(k2x–w2t)
And let x=0, y=yA+yB = 2A cos[2 (f1 – f2)t/2] cos[2 (f1 + f2)t/2]
and |f1 – f2| ≡ fbeat = = 1 / Tbeat
A(w1t)
B(w2t)
t
C(t) = A(t) + B(t)
Tbeat
Physics 207: Lecture 28, Pg 26
Exercise Superposition


The traces below show beats that occur when two different
pairs of waves are added (the time axes are the same).
For which of the two is the difference in frequency of the
original waves greater?
A. Pair 1
B. Pair 2
C. The frequency difference was the samefor both pairs of waves.
D. Need more information.
Physics 207: Lecture 28, Pg 27
Lecture 28, Dec. 8
• Assignment
 HW12, Due Friday, Dec. 12th
Physics 207: Lecture 28, Pg 29