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CCh
ap. 1
2: W
ave
1
•To describe m
echanical waves. •
To study superposition, standing waves and sound. •
To present sound as a standing longitudinal wave. •
To see that waves will interfere (add constructively and destructively).
•To study sound intensity and beats.
•To solve for frequency shifts (the D
oppler Effect). •
To examine applications of acoustics and m
usical tones.
(1)Characterization of “wave” �
1, 2, 3, 4 (2)
Wave phenom
ena �
5 – Reflection �
6 & 7 – Standing wave �
8 - Interference (3)
Something in real life
�9 – Sound & H
earing �
10 – Sound intensity �
11 – Beats �
12 – Doppler effect
onoof
“wavve”
HHow
to stu
dy …
2
•To describe m
echanical waves.
•To study superposition, standing waves and sound.
•To present sound as a standing longitudinal wave.
•To see that waves will interfere (add constructively and destructively).
•To study sound intensity and beats.
•To solve for frequency shifts (the D
oppler Effect). •
To examine applications of
acoustics and musical tones.
[2015C] 16 �
13 problems
55
The graph show
n below closely approxim
ates the displacement x of
a tuning fork as a function of time t as it is playing a single note.
What are (a) the am
plitude, (b) period, (c) frequency, and (d) angular frequency of this fork’s m
otion?
E1
. Visu
alizin
g Wave F
orm
(II)
66
An
ato
my o
f E1
(II)
HW
…
EE2
. Visu
alizin
g Sta
ndin
g Wave
7
A string is attached at one end to an electric oscillator set to vibrate
at 100 Hz. The other end is draped over a pulley and has a m
ass M
attached to it. If the string has a mass of 3 gram
s, a length of 3 meters, and there are 2 m
eters of string between theoscillator and
the pulley, what should M be so that two antinodes appear? Take g
to be
10 m/s
2. Ignore
the tension
created by
the extra
hanging string.
88
HW
…
An
ato
my o
f E2
FFirst Q
uestio
n: W
hat is W
ave?
Oscillations &
Vibrations
11
By Marcus Vicente Sauseda
From Physics Festival 2013
Try to address a single question: HOW
can we describe the wave m
otion? - Let’s start with Circular Motion �
SH
M �
Wave !
where �� = � t
y(t) = R sin ��
= R sin (�
t)
Kin. Equation of SHM
Step
1: C
ircula
r Motio
n to
SH
M
12
T = 2� / �
arc length s = R ��
s / T = R ��/ T
v = R �
�
Uniform
Circular Motion � = angular speed
SHM
� = angular frequency � T
x
R
��
y
y(t) = A sin(� t)
where �� = � t
y(t) = R cos ��
= R cos (�
t)
Kin. Equation of SHM
Step
1: C
ircula
r Motio
n to
SH
M (II)
13
arc length s = R ��
s / T = R ��/ T
v = R �
�
Uniform
Circular Motion � = angular speed
SHM
� = angular frequency � T
R
��A
T = 2� / �
x
y T
Wave on
Rope ! y
t
y(t) = A cos(� t)
SStep
2: W
ave F
orm
on
Rope
y
x 14
1
��
SStep
2: W
ave F
orm
on
Rope
y
x 21
[Q] H
ow can you describe the shape of the rope? [A
] A
, ��
��
vw
ave A
��
and TIM
E dependence…
22
WWave F
orm
on
Rope (T
ensio
n)
y
x 23
vwave = ��/ T�
vwave = ��(� /2�)�
vwave = �� f�tension
The wave function, y(x,t), gives a mathem
atical description of a wave. In this function, y is the displacem
ent of a particle at time t
and position x.
WWaves o
n a
lon
g rope u
nder ten
sion
(a)The velocity of the wave
will depend on the type and size of rope as well as the tension
(b)Wave length if f = 20 H
z
24
Example 12.2
� = linear mass density
225
A, �,�k,��
� T
Look
back
at A
nato
my o
f E1
[Q] D
irection?
Math
ematica
l Descrip
tion
of a
Wave 22
6
The w
ave function for a sinusoidal wave m
oving in the +x-direction is
y(x, t) = A sin(� t – k x), w
here k = 2π/ � , � = 2�f, ��= v T …
The w
ave function for a sinusoidal wave m
oving in the x-direction is
y(x, t) = A sin( � t + k x), w
here k = 2π/� , � = 2�f, ��= v T …
P.12-10 P.12-12
Which of the follow
ing wave functions describe a w
ave that m
oves in the –x-direction?
A. y(x,t) = A sin (–kx – ��t)
B. y(x,t) = A sin (kx + �t)
C. y(x,t) = A cos (kx + �t)
D. both B
and C.
E. all of A
, B, and C
.
Q12.2
27
Which of the follow
ing wave functions describe a w
ave that m
oves in the –x-direction?
A. y(x,t) = A sin (–kx – ��t) = A sin ( – (kx + �t) )
B. y(x,t) = A sin (kx + �t)
C. y(x,t) = A cos (kx + �t)
D. both B
and C
E. all of A
, B, and C
A12.2
28
Wave M
otion
A transverse traveling w
ave (amplitude A
, wave length ��,�and
frequency f) on a cord at t = 0 is represented by
y = A sin(2�x/�).
What w
ill be the equation for a wave traveling to the left along
the x axis as a function of x and t? [H
int] y(x, t) = ?
Exa
mple 3
31
The w
ave function for a sinusoidal wave
moving in the +x-direction is
y(x, t) = A sin(� t – k x),
where k = 2π/� , � = 2�f, ��= v T …
Wave M
otion
A transverse traveling w
ave (amplitude A
, wave length ��,�and
frequency f) on a cord at t = 0 is represented by
y = A sin(2�x/�).
What w
ill be the equation for a wave traveling to the left along
the x axis as a function of x and t? [A
] y(x, t) = A sin(2�x/� + �
�t ), where � = 2�f
Exa
mple 3
(Work
Book
)
32
The w
ave function for a sinusoidal wave
moving in the +x-direction is
y(x, t) = A sin(� t – k x),
where k = 2π/� , � = 2�f, ��= v T …
Wave M
otion
EExa
mple 4
A
transverse traveling wave on a cord is represented by
y(x, t) = 0.48 sin(0.56x + 84t)
where y and x are in m
eters and t in seconds. For this wave,
determine:
(a)the am
plitude, (b)
wavelength, frequency, velocity (m
agnitude and direction), (c)
maxim
um and m
inimum
speeds of particles of the cord, and (d)
maxim
um acceleration (m
agnitude) of the particles. [A
] …
33
The w
ave function for a sinusoidal wave
moving in the +x-direction is
y(x, t) = A sin(� t – k x),
where k = 2π/� , � = 2�f, ��= v T …
Wave M
otion
EExa
mple 4
(Work
Book
) A
transverse traveling wave on a cord is represented by
y(x, t) = 0.48 sin(0.56x + 84t)
where y and x are in m
eters and t in seconds. For this wave,
determine:
(a)the am
plitude, (b)
wavelength, frequency, velocity (m
agnitude and direction), (c)
maxim
um and m
inimum
speeds of particles of the cord, and (d)
maxim
um acceleration (m
agnitude) of the particles. [A
] A = 0.48, …
34
The w
ave function for a sinusoidal wave
moving in the +x-direction is
y(x, t) = A sin(� t – k x),
where k = 2π/� , � = 2�f, ��= v T …
Wave M
otion
A transverse w
ave pulse travels to the right along a string with
speed v = 2.0 m/s. A
t t = 0, the shape of the pulse is given by the function y = 0.45 cos(3.0x) w
here y and x are in meters and t in
seconds. For this wave, determ
ine: (a)
the wavelength, frequency, and am
plitude, (b)
maxim
um and m
inimum
speeds of particles of the string, and (c)
maxim
um and m
inimum
accelerations (magnitudes) of the
particles. [A
] …
EExa
mple 5
35
Wave M
otion
A transverse w
ave pulse travels to the right along a string with
speed v = 2.0 m/s. A
t t = 0, the shape of the pulse is given by the function y = 0.45 cos(3.0x) w
here y and x are in meters and t in
seconds. For this wave, determ
ine: (a)
the wavelength, frequency, and am
plitude, (b)
maxim
um and m
inimum
speeds of particles of the string, and (c)
maxim
um and m
inimum
accelerations (magnitudes) of the
particles. [A
] …
Wave function is:
y(x, t) = 0.45 cos(3.0x 6.0t)
Exa
mple 5
(Work
Book
)
36
The w
ave function for a sinusoidal wave
moving in the +x-direction is
y(x, t) = A sin(� t – k x),
where k = 2π/� , � = 2�f, ��= v T …
Wave M
otion
Interference
37
Seco
nd Q
uestio
n: W
hat is W
ave?
Bou
ndary C
on
ditio
ns
–W
hen a wave reflects from
a fixed end, the pulse inverts as it reflects.
–W
hen a wave reflects from
a free end, the pulse reflects without inverting..
338
http://ww
w.physicsclassroom
.com
/mm
edia/waves/fix.cfm
http://w
ww
.physicsclassroom.co
m/m
media/w
aves/free.cfm
339
Wave M
otion
AAm
azin
g Wave M
otio
n
40
Pendulum
http://ww
w.youtube.com
/watch?v=yV
kdfJ9PkRQ
Transverse and Longitudinal W
ave Motions
http://ww
w.acs.psu.edu/drussell/dem
os/waves/w
avemotion.htm
�M
echanical vibration �
Spring system
�Pendulum
system
�String fixed at both ends
�Sound (vibration of air density)
�W
ater wave �
Electromagnetic vibration �
PHYS202
�Light
Wave M
otion
441
Interferen
ces
442
Active N
oise C
an
cellatio
n
http://weloveteaching.com
/0sci208/lectures/waves/interference.htm
l http://w
ww
.gwoptics.org/ebook/interferom
eters.php
http://en.wikipedia.org/w
iki/Active_noise_co
ntrol http://94j51c5-15-1337.w
ikispaces.com/A
ctive+Noise+C
ancellation
Dan
cing w
ith In
terferences
–W
aves traveling in opposite directions on a taut string interfere with each other.
–The result is a standing wave pattern that does not m
ove on the string.
–Destructive Interference occurs where the wave displacem
ents cancel, and Constructive Interference occurs where the displacem
ents add.
–A
t the nodes (N) no m
otion occurs, and at the antinodes (A
) the amplitude of the m
otion is greatest. –
Figure on the next slide shows photographs of several standing wave patterns.
445
446
447
AAn
ato
my o
f Sta
ndin
g Wave
http://www.ic.sunysb.edu/Class/phy122ps/labs/dokuwiki/doku.php?id=phy123:lab_8 48
12 anti-nodes
SSta
ndin
g Waves o
n a
string
(a)F
T if � = 40.0 g/m and f1 =
20.0 Hz?
(b)f2 and w
avelength of second harm
onic? (c)
f3 and wavelength of second
overtone (or 3rd harm
onic)?
f3 = 3 f1
49
f2 = 2 f1
““Dan
cing” F
ire
50
Roy Montalvo
Wave M
otion
551
“Sound Magic”
http://ww
w.youtube.com
/watch?v=s9G
Bf8y0lY
0l
Informed by Taylor C
ampsey
552
In the primitive m
usical instrument shown in the figure, the
tension in the vibrating part of the wire depends on the weight of the block. If the wire has a frequency f, what will the frequency be if you replace the block with one four tim
es m
ore massive?
(a)2 f
(b)4 f (c)
2 f ©
2016 Pearson Education, Inc.
QQu
ick Q
uiz
53
LLook
back
at E
2
54
A string is attached at one end to an electric oscillator set to vibrate
at 100 Hz. The other end is draped over a pulley and has a m
ass M
attached to it. If the string has a mass of 3 gram
s, a length of 3 meters, and there are 2 m
eters of string between theoscillator and
the pulley, what should M be so that two antinodes appear? Take g
to be
10 m/s
2. Ignore
the tension
created by
the extra
hanging string.
AAm
azin
g Hea
ring
55
�20 – 200,000H
z is the approximate
range of human hearing. Below that
is infrasonic and above …. ultrasonic.
�N
ote: there are variations between anim
al species and effects on any hearing due to pressure changes.
Superhuman H
earing – Can hear far more sounds with far m
ore detail and at far
greater distances
than norm
ally hum
anly possible,
including sounds
on frequencies undetectable
by humans such as dog whistles. Superm
an is able to m
entally screen out most of these sounds to be able to function norm
ally, even in a noisy environm
ent, and can focus in on specific things, like aperson's voice or
heartbeat, even if they are in another part of the city. He can som
etimes be
shown to hear sounds on other planets, which makes it likely that his hearing is
fundamentally different from
that of a human, as sound cannot travel through
the vacuum of space. Like hum
ans and most anim
als, he is skilled at automatically
noticing his own name out of the jum
ble of several overheard conversations, m
aking him adept at quickly responding to calls of distress all over the city.
http://en.wikipedia.org/wiki/Powers_and_abilities_of_Superman (N
ov 2014)
Amplify faint or distant sounds; 100 Yard sound pick up; 40 hour battery life
A proven parabolic dish m
icrophone also known as an omnidirectional m
icrophone that am
plifies noises when pointed at a given direction. The Bionic Ear and Booster Set is a 12" parabolic dish with a record output jack to capture and record faint and distant sounds up to 100 yards away. 5 plus years in the spy m
arket and the Bionic Ear and Booster Set is still one of the most infam
ously used spy gadgets. Specifications: o
Frequency response of 100-10KHz
oGain am
p cut off at 90dB o
Am
plifies sounds by 40 dB o
Independent volume controls on headset
oO
utput jack to record sounds o
Line in jack impedance of 10V m
aximum
o
46dB Mic sensitivity
o40 hour battery life; 9V battery (not included)
556
Bionic Ear & Booster Set (Parabolic Dish)
A Bionic Sound A
mplifier M
icrophone that Picks Sound From
100 Yards Away
HHave yo
u n
oticed
?
57
This is useful for P.12-40, but not given in the form
ula sheet below, because this can be obtained from
the above equation. HH
ave yo
u n
oticed
?
58
40
�Shifts in observed frequency can be caused by m
otion of the source, the listener, or both. Exam
ples 12.10-12.13.
TTh
e Doppler E
ffect –– Eq.1
2.1
9
59
�Shifts in observed frequency can be caused by m
otion of the source, the listener, or both. Exam
ples 12.10-12.13.
HHave yo
u n
oticed
?
60
�Find the observed frequency by each of two listeners. Exam
ples 12.10-12.13.
EExa
mple 1
63
L2
L1
fS = 300 Hz
v = 340 m
/s
fS = 300 Hz
v = 340 m
/s
EExa
mple 1
(Work
Book
)
64
L2
L1
�Find the observed frequency by each of two listeners. Exam
ples 12.10-12.13.
vL = 0 m
/s
vL = 0 m
/s
EExa
mple 2
65
EExa
mple 2
(Work
Book
)
L
SA
S
B
�Shifts in observed frequency can be caused by m
otion of the source, the listener, or both. Exam
ples 12.10-12.13 and P.12-53,54,60
66
UUn
dersta
ndin
g Pro
blem
71
UUn
dersta
ndin
g Pro
blem
72
12.42: "By w
hat factor must the sound intensity be increased to
increase the sound intensity level by 12.5 dB?”
�I2 /I1 = ? = 10^{
2 /10 – 1 /10 } = 10^{ (
2 – 1 ) / 10}
�12.5 dB
= 2 –
1