PHY 113 C General Physics I
11 AM – 12:15 PM TR Olin 101
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Plan for Lecture 16:
Chapter 16 – Physics of wave motion
Review of SHM
Examples of wave motion
What determines the wave velocity
Properties of periodic waves
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PHY 113 C Fall 2013 -- Lecture 16
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PHY 113 C Fall 2013 -- Lecture 16
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Some comments on Simple Harmonic Motion
d 2x
F  kx
ma  m 2
dt
k
m=1 kg
d 2x
k
differential equation : 2   x
dt
m
general solution : x(t )  Acos(ωt  φ)
Suppose that you know:
k
2
provided
that
ω 
(in standard units)
m
x(t)=2 cos(2pt+3p)
What is k ?
k = m(2p)2
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PHY 113 C Fall 2013 -- Lecture 16
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Some comments on Simple Harmonic Motion
d 2x
F  kx
ma  m 2
dt
k
m=1 kg
d 2x
k
differential equation : 2   x
dt
m
general solution : x(t )  Acos(ωt  φ)
Suppose that you know:
k
2
provided
that
ω 
(in standard units)
m
x(t)=2 cos(2pt+3p)
What is the frequency of oscillations ?
w=2pf=2p
f=1 Hz
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PHY 113 C Fall 2013 -- Lecture 16
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Some comments on Simple Harmonic Motion
d 2x
F  kx
ma  m 2
dt
k
m=1 kg
d 2x
k
differential equation : 2   x
dt
m
general solution : x(t )  Acos(ωt  φ)
Suppose that you know:
k
2
provided
that
ω 
(in standard units)
m
x(t)=2 cos(2pt+3p)
What is amplitude of the displacement ?
xmax=2m
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PHY 113 C Fall 2013 -- Lecture 16
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Some comments on Simple Harmonic Motion
d 2x
F  kx
ma  m 2
dt
k
m=1 kg
d 2x
k
differential equation : 2   x
dt
m
general solution : x(t )  Acos(ωt  φ)
Suppose that you know:
k
2
provided
that
ω 
(in standard units)
m
x(t)=2 cos(2pt+3p)
What is the maximum velocity ?
v(t)=-2(2p) cos(2pt+3p)
vmax=4p
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PHY 113 C Fall 2013 -- Lecture 16
6
Some comments on Simple Harmonic Motion
d 2x
F  kx
ma  m 2
dt
k
m=1 kg
d 2x
k
differential equation : 2   x
dt
m
general solution : x(t )  Acos(ωt  φ)
Suppose that you know:
k
2
provided
that
ω 
(in standard units)
m
x(t)=2 cos(2pt+3p)
What is the maximum acceleration ?
a(t)=-2(2p)2 cos(2pt+3p)
amax=8p2
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PHY 113 C Fall 2013 -- Lecture 16
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Some comments on Simple Harmonic Motion
d 2x
F  kx
ma  m 2
dt
k
m=1 kg
d 2x
k
differential equation : 2   x
dt
m
general solution : x(t )  Acos(ωt  φ)
Suppose that you know:
k
2
provided
that
ω 
(in standard units)
m
x(t)=2 cos(2pt+3p)
What is the displacement at t=0.3 s ?
x(0.3)=2 cos(2p(0.3)+3p)
=2(0.309)=0.618 m
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PHY 113 C Fall 2013 -- Lecture 16
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Some comments on driven Simple Harmonic Motion
k
F(t)=F0sin(t)
d 2x
F  kx  F0sin t )
ma  m 2
dt
d 2x
k
differential equation : 2   x  F0sin t )
dt
m
F0 / m
general solution : x(t )  Acos(ωt  φ)  2
sin t )
2
w 
k
2
where ω 
m
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Webassign question (Assignment 14)
Damping is negligible for a 0.175-kg object hanging from
a light, 6.30-N/m spring. A sinusoidal force with an
amplitude of 1.70 N drives the system. At what
frequency will the force make the object vibrate with an
amplitude of 0.430 m?
F0 / m
x(t )  Acos(ωt  φ)  2
sin t )
2
w 
x(t )  X 0sin t )
F0 / m
X0 
k / m  2
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k
ω 
m
2
amplitude  X 0
F0
k
  
m mX 0
2
PHY 113 C Fall 2013 -- Lecture 16
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The wave equation
y ( x, t )
position
Wave variable
time
 y
2  y

c
2
2
t
x
2
2
What does the wave equation mean?
Examples
Mathematical solutions of wave equation and
descriptions of waves
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Example: Water waves
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Source: http://www.eng.vt.edu/fluids/msc/gallery/gall.htm
Needs more
sophistocated
analysis:
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Mechanical waves occur in continuous media. They
are described by a value (y) which changes in both
time (t) and position (x) and are characterized by a
wave velocity c: y=f(x-ct) or y=f(x+ct)
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Waves on a string:
Typical values for c:
3x108 m/s light waves
~1000 m/s wave on a string
343 m/s sound in air
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Transverse wave:
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Longitudinal wave:
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General traveling wave –
t=0
t>0
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iclicker question:
t=0
t=1 s
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t=2s
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Basic physics behind wave motion -example: transverse wave on a string with tension T
and mass per unit length m
y
qB
x
y
y
sin θ B  tan θ B 
x B
d2y
m 2  T sin θ B  T sin θ A
dt
m  μx
d2y
y 
 y
 μx 2  T 


dt
 x B x A 
1  y
y   2 y

 2
Lim 
x A  x
x 0 x  x B
2 y T 2 y

2
t
μ x 2
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The wave equation:
T
where c 
(for a string)
μ
 y
2  y
c
2
2
t
x
2
2
Solutions: y(x,t) = f (x  ct)
function of any shape
Note : Let u  x  ct
f u ) f u ) u

x
u x
 f u )  f u )  u   2 f u )

 
2
2 
x
u  x 
u 2
2
2
2
 f u )  f u )  u   2 f u ) 2

c
 
2
2 
2
t
u  t 
u
2
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2
2
PHY 113 C Fall 2013 -- Lecture 16
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iclicker question
Is it significant to write the wave equation with the
2
2
special symbols?
 y
2  y
c
2
t
x 2
A. Yes
B. No
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Examples of solutions to the wave equation:
Moving “pulse”:
y ( x, t )  y0 e
  x  ct )2
phase
factor
Periodic wave: y ( x, t )  y0 sin k x  ct )  φ )
2π
k
λ
2π
kc 
 2 πf  ω
T
 x t 

y ( x, t )  y0 sin  2 π    φ 
 λ T 

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PHY 113 C Fall 2013 -- Lecture 16
“wave vector”
not spring constant!!!
λ
c
T
22
phase (radians)
Periodic traveling waves:
 x t 

y ( x, t )  y0 sin  2 π    φ 
 λ T 

λ
c
T
velocity (m/s)
period (s); T = 1/f
wave length (m)
Amplitude
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Snapshot of periodic wave at t=t0
l
fl=c
Time plot of periodic wave at x=x0
1/f
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Combinations of waves (“superposition”)
Note that :
sin A  sin B  2 sin  12  A  B )cos 12  A  B )
 x t 

yright ( x, t )  y0 sin  2 π    φ 
 λ T 

 x t 

yleft ( x, t )  y0 sin  2 π    φ 
 λ T 

“Standing” wave:
 2πx
  2πt 
yright ( x, t )  yleft ( x, t )  2 y0 sin 
 φ  cos

 λ
  T 
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Summary of wave properties:
Wave speed c depends on the process and/or medium in which wave occurs.
Example :
Wave on a string with tension T and mass/length m : c 
T
m
p
c

Example :
Sound wave in air with pressure p and density  :
Example :
Light wave due to coupled electric and magnetic fields :
c  2.9979  108 m / s (fundamental constant)
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Example from webassign:
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phase (radians)
Periodic traveling waves:
 x t 

y ( x, t )  y0 sin  2 π    φ 
 λ T 

λ
c
T
velocity (m/s)
period (s); T = 1/f
wave length (m)
Amplitude
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Example from webassign:
 x t 

y ( x, t )  y0 sin  2 π    φ 
 λ T 

λ
c
T
 x t 

y ( x, t )
2p

y0 cos 2 π    φ 
t
T
 λ T 

maximum transverse speed :
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2p
y0
T
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