The Amazing
Fourier Transorm!
Joseph Fourier 1768 - 1830
Fourier Series
1 1
f (t) = 2 p
1 æ npt ö
sinç
÷
è
ø
n
T
n =1
N
å
Sawtooth
æ (2n - 1) pt ö
1
f (t) =
sinç
÷
ø
p n =1 2n - 1 è
T
4
N
å
Square wave
1D Vibrations
Standing waves on a vibrating string
n=1
n=2
n=3
n=4
2D Vibrations
Standing waves on a metallic plate
https://www.youtube.com/watch?v=GtiSCBXbHAg
m=5
m=4
m=3
m=2
m=1
n=1
n=2
n=3
n=4
Vibrations
Add material about:
-Failure due to vibrations
-vibrational fatigue
-acoustics in cars and planes
Structural Engineering
Nov. 7, 1940 - Tacoma Narrows bridge collapse
http://www.youtube.com/watch?v=nFzu6CNtqec
Music
Fourier analysis reveals underlying mathematical
structure in music!
Music
Fourier analysis explains why
certain notes sound good
together!
When notes are played
simultaneously, overtones
“harmonize”.
Image Compression and Filtering
FFT is often used for finger
print analysis.
Complex Systems: Turbulence
The Fourier transform reveals underlying
mathematical structure in complex systems.
Complex Systems: Turbulence
The velocity at a given point
in the flow fluctuates
stochastically.
Complex Systems: Turbulence
Kolmogorav’s Power Law:
U(w) µ w -5/ 3
2
Power spectral density of
velocity obeys a power law.
Complex Systems: The Stock Market
Complex financial
systems exhibit
stochastic “turbulent”
behavior.
Complex Systems: The Stock Market
Fourier analysis reveals
underlying structure in
complex systems.
“Pink noise”
f (w) µ w -2
2
PSD obeys a power law
Cosmology
Microwave radiation left over from the Big bang still persists today.
Measuring this radiation tells us a lot about the universe!
Planck microwave telescope
$800 Million Experiment!
Map of the radiation taken by
Planck.
Cosmology
At a glance, the angular
distribution of the radiation
appears to be random...
Map of the radiation taken by
Planck.
Taking a special Fourier
Transform called a “multipole
expansion” reveals underlying
structure.