Spring 2016
Ch4. Coupled Oscillations
6. Wave Motion as Coupled Oscillations
3/3/2016
The Wave Equation of a String
2
y
M2
1
T
y  x, t 
M1
T
δx
x
x+δx
x
2
2 y T 2 y

y
2

c
2
2
t
 x
x 2
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2
Wave Motion as Coupled Oscillations
 Oscillation of infinite no. of coupled particles (lattice/medium):
2
2 y

y
2
c
2
t
x 2
y ( x, t )  f ( x  ct )
yn  An cos(nt  kx)
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Analyzing a Wave
• The amplitude is the maximum displacement of the
medium from its equilibrium position.
• The wavelength () is the minimum distance between
two points which are in phase.
• The frequency (ƒ) is the number of complete oscillations
made in one second.
Unit : Hz
• The period (T) is the time taken for one complete
oscillation. It is related to frequency by T=1/f
Unit : s
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Types of Waves
Mechanical waves in string
(transverse wave)
y  y ( x, t )
2
2 y
2  y
c
2
t
x 2
Sound waves in air
(longitudinal wave)
Gas displacement due to disturbance
   ( x, t )
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2
 2


2
c
2
t
x 2
c: sound speed
Types of Waves
W
A
V
E
S
M
e
c
h
a
n
ic
a
lw
a
ves
T
ra
n
sve
rsew
a
ve
s
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E
le
c
tro
m
a
gn
e
ticw
a
ves
L
o
n
g
itu
d
in
a
lw
a
ve
s
T
ra
n
sve
rsew
a
ve
s
Waves by Simple Harmonic Motions
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Plane Wave Representation
Amplitude
Phase
Angular frequency
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Wave Number
Fix time
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Phase Velocity
Follow
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@
Wave Propagation
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Spring 2016
Ch6 Continuous Wave Systems
and Fourier Analysis
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Coupled Oscillations of a Loaded String
y
T
a
x
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A Complete Spectrum of Normal Modes
It is possible to take any form of profile of the string, described by a function of x
between x=0 and x=L and analyze It into an infinite series of sine functions.
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What Can We Learn?
We desire a measure of the frequencies present in a wave. This will
lead to a definition of the term, the "spectrum.“
Plane waves have only
one frequency, .
This light wave has many
frequencies – wave group.
The frequency increases in
time (from red to blue).
It will be nice if our measure also tells us when each frequency occurs.
3/3/2016
Sums of Sinusoids
Consider the sum of two sine waves (i.e., harmonic
waves) of different frequencies:
The resulting wave is periodic, but not harmonic.
Most waves are anharmonic.
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Fourier
Decomposing
Functions
Here, we write a
square wave as
a sum of sine waves.
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Any Function Can Be Written as the
Sum of an Even and an Odd Function
E ( x)  [ f ( x)  f (  x)] / 2
O( x)  [ f ( x)  f (  x)] / 2

f ( x)  E ( x)  O( x)
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Even Function – Fourier Cosine Series
Because cos(mt) is an even function (for all m), we can write an even
function, f(t), as:
f(t) 

F


1
m
cos(mt)
m 0
where the set {Fm; m=0, 1, … } is a set of coefficients that define the
series.
And where we’ll only worry about the function f(t) over the interval
(–π,π).
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The Kronecker Delta Function
 m,n
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1 if m  n

0 if m  n
Finding the coefficients in a Fourier Cosine Series


Fourier Cosine Series:
f (t ) 
1

Fm cos(mt )
m0
To find Fm, multiply each side by cos(m’t), where m’ is another integer, and integrate:






m0

1


Fm   m ,m '
 only the m’ = m term contributes
m0

Dropping the ‘ from the m:
Fm 
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

f (t ) cos(m ' t ) dt 

Fm cos(mt) cos(m' t) dt
 if m  m '
 
   m,m '
 0 if m  m '
cos(mt ) cos(m ' t ) dt


So:
1
f (t) cos(m' t) dt 

But:




f (t ) cos(mt ) dt
 yields the
coefficients for
any f(t)!
Fourier Sine Series
Because sin(mt) is an odd function (for all m), we can write
any odd function, f(t), as:


f (t) 
1

Fmsin(mt)
m 0
where the set {F’m; m=0, 1, … } is a set of coefficients that define the
series.
where we’ll only worry about the function f(t) over the interval (–π,π).
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Finding the coefficients in a Fourier Sine Series


1
f (t) 
Fourier Sine Series:

Fmsin(mt)
m 0
To find Fm, multiply each side by sin(m’t), where m’ is another integer, and
integrate:


1 
 f (t ) sin(m ' t ) dt    Fm sin(mt ) sin(m ' t ) dt




But:
So:
m0

 if m  m'
sin(mt) sin(m' t) dt  
0 if m  m'
   m ,m'





f (t) sin(m' t) dt


Dropping the ‘ from the m:
1

Fm  m ,m'  only the m’ = m term contributes
m 0

Fm 


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f (t) sin(mt) dt  yields the coefficients
for any f(t)!
Fourier Series


f (t) 
1



Fm cos(mt) 
1

Fmsin(mt)
m 0
m 0
even component
odd component
where
Fm 
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
f (t) cos(mt) dt
and
Fm 

f (t) sin(mt) dt
The Coefficients of a Fourier Series
Fm vs. m
We really need two such plots, one for the cosine series and another
for the sine series.
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Discrete Fourier Series vs.
Continuous Fourier Transform
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The Fourier Transform
Consider the Fourier coefficients. Let’s define a function F(m)
that incorporates both the cosine and sine series coefficients, with the sine
series coefficients distinguished by making them the imaginary component:
F(m)  Fm – i F’m =

f (t) cos(mt) dt  i

f (t) sin(mt) dt
Let’s now allow f(t) to range from – to , so we’ll have to integrate from –
to , and let’s redefine m to be the “frequency,” which we’ll now call w:

F ( ) 

f (t ) exp(i t ) dt
The Fourier
Transform

F(w) is called the Fourier Transform of f(t). It contains equivalent information
to that in f(t). We say that f(t) lives in the “time domain,” and F(w) lives in the
“frequency domain.” F(w) is just another way of looking at a function or wave.
3/3/2016
What Can We Learn?
We desire a measure of the frequencies present in a wave. This will
lead to a definition of the term, the "spectrum.“
Plane waves have only
one frequency, .
This light wave has many
frequencies – wave group.
The frequency increases in
time (from red to blue).
It will be nice if our measure also tells us when each frequency occurs.
3/3/2016
The Inverse Fourier Transform
The Fourier Transform takes us from f(t) to F(). How about going
back?
Recall our formula for the Fourier Series of f(t):
f(t) 
1


F
m
m 0
cos(mt) 
1


 F'
m
sin(mt)
m 0
Now transform the sums to integrals from – to , and again replace
with F(). Remembering the fact that we introduced a factor of i (and
including a factor of 2 that just crops up), we have:
1
f (t ) 
2
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


F ( ) exp(i t ) d 
Inverse
Fourier
Transform
The Fourier Transform and its Inverse
The Fourier Transform and its Inverse:

F ( ) 
 f (t ) exp(it ) dt
FourierTransform

f (t )

1
2

 F ( ) exp(it )d
Inverse Fourier Transform

So we can transform to the frequency domain and back.
Interestingly, these functions are very similar.
There are different definitions of these transforms. The 2π can
occur in several places, and is generally written in the above form.
3/3/2016
Fourier Transform Notation
There are several ways to denote the Fourier transform of a
function.
If the function is labeled by a lower-case letter, such as f,
we can write:
f(t)  F()
If the function is labeled by an upper-case letter, such as E, we can
write:
E (t )  {E (t )}
or:
3/3/2016
E (t )  E ( )
Fourier Transform Symmetry Properties
Expanding the Fourier transform of a function, f(t):


F ( ) 
[Re{ f (t )}  i Im{ f (t )}] [cos( t )  i sin( t )] dt

Expanding further:
= 0 if Re or Im{f(t)} is odd


F ( ) 


Re{ f (t )} cos( t ) dt 

= 0 if Re or Im{f(t)} is even



Im{ f (t )} sin( t) dt


 i  Im{ f (t )} cos( t ) dt  i  Re{ f (t )} sin( t) dt


Even functions of 
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Re{F()}
Odd functions of 
Im{F()}
The Spectrum
We define the spectrum of a wave E(t) to be:
{E (t )}
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2
Example: the Fourier Transform of a
decaying exponential: exp(-at) (t > 0)

F (    exp(  at ) exp( it ) dt
0


0
0
  exp(at  it )dt   exp( [ a  i t ) dt
1
1

exp([ a  i t ) 0 
[exp()  exp(0)]
a  i
a  i
1

[0  1]
a  i
1

a  i

1
F (   i
  ia
A complex Lorentzian!
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Complex Lorentzian:
Intensity and Phase
a
Real
component
0
0
Imaginary
component

Real part
Imag part
1
1 a  i
a

L( ) 

 2
2 i 2
2
a  i
a  i a  i
a 
a 
Intensity ( ) 
1
2
2
a 
Im[L( )] 
Phase( )  arctan 
   arctan( / a)
Re[L( )] 
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The Meaning of Fourier Transform
Ex:  pulse-driven dipole oscillators
“Ringing forks”
E(t)
P(t)
 pulse excitation
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t
t
Projection between the conjugate quantities
Example: the Fourier Transform of a
rectangle function: rect(t)
1/ 2
1
F ( )   exp( i t )dt 
[exp( i t )]1/1/2 2
i
1/ 2
1
[exp(i / 2)  exp(i
i
 exp(i / 2)  exp(i sin(



2i


F (   sinc(
Imaginary
Component = 0
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More About Sinc(x)
Sinc(x/2)
transform
function.
is
of
the Fourier
a rectangle
Sinc2(x/2)
transform
function.
is
of
the
a
Fourier
triangle
Sinc2(ax) is the diffraction
pattern from a slit.
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The Fourier Transform of the Triangle
function, L(t), is sinc2(w/2)

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The Convolution
The convolution allows one function to smear or broaden another.

f (t )  g (t ) 

f ( x) g (t  x) dx





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f (t – x) g ( x) dx
changing variables:
x  t-x
rect(x) * rect(x) = L(x)
We can perform
convolution visually.
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The Fourier Transform of the Triangle
function, L(t), is sinc2(w/2)

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Example
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The Convolution Theorem
The Convolution Theorem turns a convolution into the inverse FT of
the product of the Fourier Transforms:
{f (t )  g (t )} = F ( w
) G ( )


{ f (t ) * g (t )}     f ( x) g (t – x) dx  exp( i t ) dt
  

Proof:










f ( x)   g (t  x) exp(– i t ) dt  dx
 

f ( x){G (  exp(–i x)} dx



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

f ( x) exp(–i x) dxG (   F ( G ( 
The Autocorrelation

The autoconvolution of a function f(x) is given by: f  f 

f (u ) f ( x  u )du

Suppose, however, that prior to multiplication and integration we do not reverse one
of the two component factors; then we have the integral: 

f (u ) f (u  x)du

which may be denoted by f f. A single value of f f is represented by:


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
f (u ) f * (u  x)du
The Fourier Transform of the Triangle
function, L(t), is sinc2(w/2)

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The Fourier Transform of the autocorrelation


2
   f (u ) f *(u  x) du    f ( x)




   f (u ) f *(u  x) du  

 










 exp(ikx) 

f (u ) f *(u  x) du dx

*

f (u )   exp(ikx) f (u  x)dx  du

 
*

f (u )   exp(iky ) f (u  y )dy  du

 




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f (u ) exp(iku ) du F *(k )  F (k ) F *(k )  F (k )
2
Examples
The Autocorrelation Theorem can be applied to a light
wave field, yielding important result:


2
   E (t ) E *(t   dt   {E (t )}
 

 E ( 
2
= the spectrum!
Remarkably, the Fourier transform of a light-wave field’s autocorrelation is its
spectrum!
This relation yields an alternative technique for measuring a light wave’s
spectrum.
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The Dirac Delta Function
Unlike the Kronecker delta-function, which is a function
of two integers, the Dirac delta function is a function of
a real variable, t.
 if t  0
 (t )  
 0 if t  0
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Convolution with a Delta Function
f (t )   t  a ) 


f (t  u )  (u – a) du

 f (t  a )
Convolution with a delta function simply centers the function on the
delta-function.
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Dirac -function Properties

  (t ) dt  1


  (t  a) f (t ) dt 
f (a )


 exp(it ) dt  2  ( 


 exp[i(   ')t ]

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dt  2  (   '
The Fourier transform of exp(i0t)


F{exp(i0t)} =
exp( i 0 t) exp( i  t ) dt


=

exp( i [   0 ]t) dt

= 2π  (   0 )
F{exp(i0t)}
0
0
The function exp(i0t) is the essential component of
Fourier analysis. It is a pure frequency.
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
The Fourier transform of cos(0t)

F{cos(0t)}
1
2
=
1
2
=
=
=
 cos(
0
t ) exp(i  t ) dt


 exp(i 
0
t )  exp(i 0 t )  exp(i  t ) dt



1
2
exp(i [  0 ] t ) dt 



exp(i [  0 ] t ) dt

  (   0 )    (   0 )
{cos( 0 t)}

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-0
0
+0
Some Fourier Transform Theorems
The Fourier transform of a scaled function
{ f (at )}  F ( /a) / a
The Fourier Transform of a sum of two functions
{ f (t )  g (t )} 
{ f (t )}  {g (t )}
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Shift Theorem
 f (t  a)  exp(i a) F ( )
Intensity & Phase of Fourier Transform
A laser pulse has the time-domain electric field:
E (t) = Re {
I(t)1/ 2
exp [ i  t – i  (t) ] }
Intensity
0
Phase
(neglecting the
negative-frequency
component)
Equivalently, vs. frequency:
~
E (  ) = Re {
S(0)1/ 2
Spectrum
exp [- i j ( –  0) ] }
Spectral
Phase
Knowledge of the intensity and phase or the spectrum and spectral phase
is
sufficient to determine the light wave.
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