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Chap. 15: Wave
1
I. Outlook
II. What is wave?
III.Shape & Examples
IV. Equation of motion – Wave equations
V. Examples
Wave Motion
Outlook Translational and Rotational Motions
with Several physics quantities
Energy (E)
Momentum (p)
Angular momentum (L)
With Conservation laws
Conservation of energy
Conservation of linear momentum
Conservation of angular momentum
Wave Motion
2
Wave Motion
What is Wave? Various Types of Waves
3
Mechanical vibration
Spring system
String fixed at both ends
Sound (vibration of air density)
Water wave
Electromagnetic vibration PHYS208
Light
Wave Motion
Wave Motion
7
Pendulum
http://www.youtube.com/watch?v=yVkdfJ9PkRQ
Transverse and Longitudinal Wave Motions
http://www.acs.psu.edu/drussell/demos/waves/wavemotion.htm
Wave Motion
Goal of My Lectures
Try to address a single question:
• HOW can we describe the wave motion?
Let’s start with SHM!
8
Wave Motion
x
x = R0 cos q
where q = w t
R0
x(t) = R0 cos (w t)
q
Starts here
Kin. Equation of SHM
x(t) = R0 cos (w t + f )
f
“phase angle”
Review: Circular Motion to SHM Simple Harmonic Oscillator (SHO)
9
Figure 15.4
and TIME dependence…
22
cos ),(
tvx
k
Atxy
w
co s )( tAty w
2
cos )(
xAxy
22
sin ),(
tvx
k
Atxy
w
11
Wave Motion
x
y
2
s in )0,(
xAtxy
2
cos
2
2
sin
2
2
sin
4
22 sin )
4(
xA
xA
vTxA
TvxA
Tt,xy
w
16
A
– A
wave speed v = /T
or = v T
Mathematical Description of a Wave
– The wave function, y(x,t), gives a
mathematical description of a
wave. In this function, y is the
displacement of a particle at time t
and position x.
– The wave function for a sinusoidal
wave moving in the +x-direction is
y(x,t) = Acos(kx – wt), where k =
2π/ is called the wave number.
– Figure 15.8 at the right illustrates a
sinusoidal wave.
17
Mathematical Description of a Wave
– The wave function, y(x,t), gives a
mathematical description of a
wave. In this function, y is the
displacement of a particle at time t
and position x.
– The wave function for a sinusoidal
wave moving in the +x-direction is
y(x,t) = Acos(kx – wt), where k =
2π/ is called the wave number.
– Figure 15.8 at the right illustrates a
sinusoidal wave.
18
The wave function for a sinusoidal wave
moving in the +x-direction is
y(x, t) = A cos(k x – w t),
where k = 2π/ , w = 2f, = v T …
Mathematical Description of a Wave
– The wave function for a sinusoidal
wave moving in the x-direction is
y(x,t) = Acos(kx + wt), where k =
2π/ is called the wave number.
19
The wave function for a sinusoidal wave
moving in the x-direction is
y(x, t) = A cos(k x + w t),
where k = 2π/ , w = 2f, = v T …
Wave Motion
A transverse traveling wave (amplitude A, wave length , and
frequency f) on a cord at t = 0 is represented by
y = A sin(2x/ + f).
Here f is a constant phase factor.
(a) What will be the equation for a wave traveling to the left
along the x axis as a function of x and t?
[Hint] y(x, t) = ? (find …)
(b) What is its maximum acceleration of particles on the
cord?
[Hint] ay(x, t) = ?
Example 1
20
The wave function for a sinusoidal wave
moving in the +x-direction is
y(x, t) = A cos(k x – w t),
where k = 2π/ , w = 2f, = v T …
Wave Motion
A transverse traveling wave (amplitude A, wave length , and
frequency f) on a cord at t = 0 is represented by
y = A sin(2x/ + f).
Here f is a constant phase factor.
(a) What will be the equation for a wave traveling to the left
along the x axis as a function of x and t?
[Hint] y(x, t) = ? (find w)
(b) What is its maximum acceleration of particles on the
cord?
[Hint] ay(x, t) = ?
Example 1
21
The wave function for a sinusoidal wave
moving in the +x-direction is
y(x, t) = A cos(k x – w t),
where k = 2π/ , w = 2f, = v T …
Wave Motion
A transverse traveling wave (amplitude A, wave length , and
frequency f) on a cord at t = 0 is represented by
y = A sin(2x/ + f).
Here f is a constant phase factor.
(a) What will 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(2x/ + w t + f), where w = 2f
(b) What is its maximum acceleration of particles on the
cord?
[Hint] ay(x, t) max ay?
Example 1
22
Wave Motion
A transverse traveling wave (amplitude A, wave length , and
frequency f) on a cord at t = 0 is represented by
y = A sin(2x/ + f).
Here f is a constant phase factor.
(a) What will 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(2x/ + w t + f), where w = 2f
(b) What is its maximum acceleration of particles on the
cord?
[A] ay(x, t) = –Aw2 sin(2x/ + w t + f) aymax= …
Example 1
23
Wave Motion
A transverse traveling wave (amplitude A, wave length , and
frequency f) on a cord at t = 0 is represented by
y = A sin(2x/ + f).
Here f is a constant phase factor.
(a) What will 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(2x/ + w t + f), where w = 2f
(b) What is its maximum acceleration of particles on the
cord?
[A] ay(x, t) = –Aw2 sin(2x/ + w t + f) aymax=Aw2
Example 1
24
Wave Motion
Example 2 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 meters and t in seconds. For this wave,
determine:
(a) the amplitude,
(b) wavelength, frequency, velocity (magnitude and direction),
(c) maximum and minimum speeds of particles of the cord, and
(d) maximum acceleration (magnitude) of the particles.
[A] …
25
The wave function for a sinusoidal wave
moving in the +x-direction is
y(x, t) = A cos(k x – w t),
where k = 2π/ , w = 2f, = v T …
Wave Motion
A transverse wave pulse travels to the right along a string with
speed v = 2.0 m/s. At t = 0, the shape of the pulse is given by the
function y = 0.45 cos(3.0x + 1.2) where y and x are in meters and t
in seconds. For this wave, determine:
(a) the wavelength, frequency, and amplitude,
(b) maximum and minimum speeds of particles of the string, and
(c) maximum and minimum accelerations (magnitudes) of the
particles.
[A] …
Example 3
26
The wave function for a sinusoidal wave
moving in the +x-direction is
y(x, t) = A cos(k x – w t),
where k = 2π/ , w = 2f, = v T …
If you double the wavelength of a wave on a string,
what happens to the wave speed v and the wave
frequency f?
A. v is doubled and f is doubled.
B. v is doubled and f is unchanged.
C. v is unchanged and f is halved.
D. v is unchanged and f is doubled.
E. v is halved and f is unchanged.
Q15.1
The wave function for a sinusoidal wave
moving in the +x-direction is
y(x, t) = A cos(k x – w t),
where k = 2π/ , w = 2f, = v T = v / f …
If you double the wavelength of a wave on a string,
what happens to the wave speed v and the wave
frequency f?
A. v is doubled and f is doubled.
B. v is doubled and f is unchanged.
C. v is unchanged and f is halved.
D. v is unchanged and f is doubled.
E. v is halved and f is unchanged.
A15.1
Which of the following wave functions describe a wave
that moves in the –x-direction?
A. y(x,t) = A sin (–kx – wt)
B. y(x,t) = A sin (kx + wt)
C. y(x,t) = A cos (kx + wt)
D. both B. and C.
E. all of A., B., and C.
Q15.2
Which of the following wave functions describe a wave
that moves in the –x-direction?
A. y(x,t) = A sin (–kx – wt)
B. y(x,t) = A sin (kx + wt)
C. y(x,t) = A cos (kx + wt)
D. both B. and C.
E. all of A., B., and C.
A15.2
Wave Motion
[Q] The figure shows the wave shape of a
sinusoidal wave traveling to the right at two
instants of time. Find the mathematical
representation of the wave?
[A] …
Example 5
31
The wave function for a sinusoidal wave
moving in the +x-direction is
y(x, t) = A cos(k x – w t),
where k = 2π/ , w = 2f, = v T …
txkAt,xy
k,.A
v,.A
tkxAt,xy
tkxAt,xy
v,k
tkxAt,xy
cos )(
, cm 53
s 3 /cm 4 cm, 6 cm 53
cos )(
right the to Moveing
cos )(
90
2
2 where
sin )(
w
w
w
w
f
w
fw
Wave Motion
Example 5
4
= 6 cm
32
v = 4 cm / 3.0 s
The wave function for a sinusoidal wave
moving in the +x-direction is
y(x, t) = A cos(k x – w t),
where k = 2π/ , w = 2f, = v T …
cos
cos
sin sin
f = ?
txkAt,xy
k,.A
v,.A
tkxAt,xy
tkxAt,xy
v,k
tkxAt,xy
cos )(
, cm 53
s 3 /cm 4 cm, 6 cm 53
cos )(
right the to Moveing
cos )(
90
2
2 where
sin )(
w
w
w
w
f
w
fw
Wave Motion
4
= 6 cm
Example 5
33
cos
cos
sin sin
+ or ?
v = 4 cm / 3.0 s
txkAt,xy
k,.A
v,.A
tkxAt,xy
tkxAt,xy
v,k
tkxAt,xy
cos )(
, cm 53
s 3 /cm 4 cm, 6 cm 53
cos )(
right the to Moveing
cos )(
90
2
2 where
sin )(
w
w
w
w
f
w
fw
Wave Motion
4
= 6 cm
Example 5
34
v = 4 cm / 3.0 s
A = ? k = ? w = ?
Wave Motion
4
= 6 cm
Example 5
35
txkAt,xy
k,.A
v,.A
tkxAt,xy
tkxAt,xy
v,k
tkxAt,xy
cos )(
, cm 53
s 3 /cm 4 cm, 6 cm 53
cos )(
right the to Moveing
cos )(
90
2
2 where
sin )(
w
w
w
w
f
w
fw
Wave Motion
Math & Physics
– Equation of Motions –
?) (2
sin )(
d
d ) cos( )(
)( 2
1 )(
2
2
2
2
00
tvxAt,xy
xt
xtAtx
m/Fatatvxtx
ww
36
Wave Motion
Math
2
2
2
2
2
2
2
2
2
2
2
2
2
22
2
2
sin
[RH]
sin
[LH]
2
2 where
cos
) (
2 sin )(
x
y
kt
y
tkxAkx
y
tkxAt
y
v,k
,tkxAt
y
x
yv
t
ytvxAt,xy
w
w
ww
w
ww
Differential Equation
Solution of D.Eq.
2nd derivative?
37
Wave Motion
Math
2
2
2
2
2
2
2
2
2
2
2
2
2
22
2
2
sin
[RH]
sin
[LH]
2
2 where
cos
) (
2 sin )(
x
y
kt
y
tkxAkx
y
tkxAt
y
v,k
,tkxAt
y
x
yv
t
ytvxAt,xy
w
w
ww
w
ww
Differential Equation
Solution of D.Eq.
Where is Physics?
38
Wave Motion
2
2
2
2
2
2
2
2
2
2
2
2
2
22
2
2
sin
[RH]
sin
[LH]
2
2 where
cos
) (
2 sin )(
x
y
kt
y
tkxAkx
y
tkxAt
y
v,k
,tkxAt
y
x
yv
t
ytvxAt,xy
w
w
ww
w
ww
Math & Physics – Equation of Motions –
ay = –w2 y (You have seen this!)
39
2
2
2
vk
w
Boundary Conditions
– When a wave reflects from a fixed end, the pulse inverts as it reflects. See Fig. 15.19(a) at the right.
– When a wave reflects from a free end, the pulse reflects without inverting. See Fig. 15.19(b) at the right.
42
Dancing with Interferences – Waves traveling in opposite directions on a taut string
interfere with each other.
– The result is a standing wave pattern that does not move on the string.
– Destructive Interference occurs where the wave displacements cancel, and Constructive Interference occurs where the displacements add.
– At the nodes (N) no motion occurs, and at the antinodes (A) the amplitude of the motion is greatest.
– Figure 15.23 on the next slide shows photographs of several standing wave patterns.
43
Standing Wave
Wave Motion
http://www.ic.sunysb.edu/Class/phy122ps/labs/dokuwiki/doku.php?id=phy123:lab_8
44
Wave Motion
50
“Sound Magic”
http://www.youtube.com/watch?v=s9GBf8y0lY0l
Informed by Taylor Campsey
54
4
interference
http://weloveteaching.com/0sci208/lectures/w
aves/interference.html http://www.gwoptics.org/ebook/interferome
ters.php
55
4
http://en.wikipedia.org/wiki/Acti
ve_noise_control
http://94j51c5-15-
1337.wikispaces.com/Active+Nois
e+Cancellation
Active Noise Cancellation
Wave Motion
Pulse on Rope “Periodic Motion” y
x
[Q] How can you describe the shape of the rope?
[A]
59
Wave Motion
Pulse on Rope
[Q] How can you describe the shape of the rope?
[A] Well, I use w (because it is a periodic motion)!
[Q] Anything else?
“Periodic Motion” y
x
60
Wave Motion
Pulse on Rope
[Q] Anything else?
[A] The rope can be treated as a group of particles,
each being a SHO in y axis.
Rope = Group of particles
“Continuum of SHOs” y
x
y(t) = ? 61
Wave Motion
Pulse on Rope
[Q] Anything else?
[A] The rope can be treated as a group of particles,
each being a SHO in y axis.
Rope = Group of particles
“Continuum of SHOs” y
x
y(t) = A cos (w t + f )
A
-A
62
Wave Motion
Pulse on Rope
[Q] Can we describe y for all SHO’s?
[A] Hmmm … Not so easy … Let’s go slowly.
Let’s consider a shape of rope at t = 0.
Rope = Group of particles
“Continuum of SHOs” y
x
A
-A
63
Wave Motion
Shape y
x
A
– A
2
s in )(
xAxy
Wave number is
“number of waves
in unit length”:
k = 2/
So, how many waves
in unit of 2?
0 1 m
[Q] How can you describe the shape of the rope?
[A]
64
Wave Motion
Shape y
x
A
– A
2
s in )(
xAxy
Wave number is
“number of waves
in unit length”:
k = 2/
So, how many waves
in unit of 2?: 1.5
0 1 m
[Q] How can you describe the shape of the rope?
[A] Use T, (or k), and A
[Q] Anything else? 65
Wave Motion
Shape y
x
A
– A
and TIME dependence…
2
s in )(
xAxy
Wave number is
“number of waves
in unit length”:
k = 2/
0 1 m
66
Wave Motion
x
A
– A
2
s in )0,(
xAtxy
1
2 3
4
¼ of one cycle ¼ T
Motion of wave by /4
Motion of SHO
y
wave speed v = (/4)/(T/4)
v = /T or = v T
Shapes
68
Wave Motion
x
y
A
– A
2
s in )0,(
xAtxy
[y] SHM angular frequency
[x] Motion with a constant velocity
Two Components
Bottom line is … 69
Wave Motion
x
y
A
– A
2
s in )0,(
xAtxy
22
sin ),(
tvx
k
Atxy
w
42
2
xx
44
2
4
22
Tvx
Tvx
The shape of the rope (wave)
moves to +x direction.
70
Wave Motion
Wave Motion
I. Outlook
II. What is wave?
III.Kinematics & Examples
IV. Equation of motion – Wave equations
V. More Examples
Sections 1,2,4,5,
72
Wave Motion
x
y
A
– A
2
s in )0,(
xAtxy
[y] SHM angular frequency
[x] Wave propagation:
moving with a constant velocity
22
sin ),(
tvxAtxy
w
Visualization
73
Wave Motion
Math & Physics
– Equation of Motions –
?) (2
sin )(
d
d ) cos( )(
)( 2
1 )(
2
2
2
2
00
tvxAt,xy
xt
xtAtx
m/Fatatvxtx
ww
74
Wave Motion
Finding Wave Equations
- Transverse Wave on Rope -
Consider a segment (mass m)
of the rope under FT.
y
x
75
Wave Motion
Finding Wave Equation
Consider a segment (mass m)
of the rope under FT.
Look at the vertical (y) motion.
1
2
FT
FT
m
y
x
76
Wave Motion
1
2
FT
FT
m
x
x
y
F
t
y
x
F
t
y
m
F
t
y
m
F
t
y
m
FF
t
y
m
FF
t
y
m
Fa
yy
y
y
SLOPE) of e(differenc
)SLOPE(SLOPE
)(
sinsin
T
2
2
T
2
2
12T
2
2
12T
2
2
1T2T
2
2
12
2
2
:velocity Wave
:Equation Wave
T
2
2
T
2
2
Fv
x
yF
t
y
Finding Wave Equation y
x
78
Wave Motion
Finding Wave Equation
1
2
FT
FT
m
:velocity Wave
:Equation Wave
T
2
2
T
2
2
Fv
x
yF
t
y
y
x Consider a segment (mass m)
of the rope under FT.
Look at the vertical (y) motion.
79
Wave Motion
[27] Determine if the function y = A sin (k x – w t)
is a solution of the wave equation.
[A] …
Example 4
80
Wave Motion
Math
2
2
2
2
2
2
2
2
2
2
2
2
2
22
2
2
sin
[RH]
sin
[LH]
2
2 where
cos
) (
2 sin )(
x
y
kt
y
tkxAkx
y
tkxAt
y
v,k
,tkxAt
y
x
yv
t
ytvxAt,xy
w
w
ww
w
ww
Differential Equation
Solution of D.Eq.
Where is Physics?
81
Wave Motion
2
2
2
2
2
2
2
2
2
2
2
2
2
22
2
2
sin
[RH]
sin
[LH]
2
2 where
cos
) (
2 sin )(
x
y
kt
y
tkxAkx
y
tkxAt
y
v,k
,tkxAt
y
x
yv
t
ytvxAt,xy
w
w
ww
w
ww
Math & Physics – Equation of Motions –
ay = –w2 y (You have seen this!)
82