Mayda M. Velasco Winter Classical Mechanics: Lecture #20

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

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

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)

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

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

Orthogonal set of Sinusoidal Functions

Fourier Decomposition

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

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

Square wave, f(x)=1

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

Harmonics

Amplitudes and Phase Angles harmonic amplitude phase angle

Complex form

Complex Form of Fourier Series

Fourier cosine series

Comparison of sine and cosine series

sawtooth wave triangle wave

Full range Fourier series