Spring 2016
Ch4. Coupled Oscillations
6. Wave Motion as Coupled Oscillations
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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
Ch5 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.
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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.
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