1. Mayda M. Velasco Winter 2015-2016 Classical Mechanics: 330-2 Lecture #20

2. Last lecture Wave Equation: Plane waves: Ex. photons Spherical waves: Ex. Electron

3. A periodic sequence T2T3T t f(t)f(t) The Mathematic Formulation of Fourier Method Any function that satisfies: where T is a constant T is the period of the function  Decompose a periodic input signal into primitive periodic components

4. Even and Odd Functions A function f(x) is even when f(x) = f(-x) if f(x) = - f(-x), the function is an odd function. An even function x f(x) x An odd function Ex: cos(x) Ex: sin(x)

5. Orthogonal Functions Call a set of functions  k orthogonal on an interval a < t < b if: Is an orthogonalset orthogonalset

6. Fourier Method Const. Part Even Part Odd Part T is a period of all the above signals Let  0 =2  /T

7. Orthogonal set of Sinusoidal Functions

8. Fourier Decomposition

9. Example (Square Wave)  2  3  4  5  -- -2  -3  -4  -5  -6  f(t)f(t) 1

10.  2  3  4  5  -- -2  -3  -4  -5  -6  f(t)f(t) 1

11. Square wave, f(x)=1

17. Harmonics Define, called the fundamental angular frequency. Define, called the n-th harmonic of the periodic function.

18. Harmonics

19. Amplitudes and Phase Angles harmonic amplitude phase angle

20. Complex form

21. Complex Form of Fourier Series

24. Fourier cosine series

29. Comparison of sine and cosine series

30. sawtooth wave triangle wave

31. Full range Fourier series