The Amazing Fourier Transorm! Joseph Fourier Fourier Series Sawtooth Square wave 1D Vibrations Standing waves on a vibrating string n = 1 n = 2 n = 3n = 4 2D Vibrations Standing waves on a metallic plate n = 1 n = 2 n = 3n = 4 m = 1 m = 2 m = 3 m = 4 m = 5 https://www.youtube.com/watch?v=GtiSCBXbHAg Structural Engineering Nov. 7, Tacoma Narrows bridge collapse 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 Kolmogoravs Power Law: 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 PSD obeys a power law Cosmology Planck microwave telescope $800 Million Experiment! Microwave radiation left over from the Big bang still persists today. Measuring this radiation tells us a lot about the universe! 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.