1. Wave Physics PHYS 2023 Tim Freegarde

2. Fourier transforms Uses of Fourier transforms: Reveal which frequencies/wavenumbers are present identification or analysis system performance definition Energy/power/intensity calculations orthogonality means component powers may simply be added Propagation in dispersive systems determine propagation of individual components, and add group velocity Fraunhofer diffraction Bandwidth theorem / Heisenburg uncertainty principle Convolution theorem

3. 3 Fourier analysis any function may be written as a superposition coefficients or etc. are found using the property of orthogonality multiply waveform by function whose coefficient is required integrate product over range of and normalize complications...: periodic and non-periodic waveforms complex exponential basis functions limits or factors of use, or as appropriate

4. 4 Fourier transform variations* reference functions:signs:integration limits:scale factors:function periodicity:terminology: transformanalysissynthesis discrete transform continuous transform * no correlations between rows – ie transform doesn’t mean scale factor = 1 etc.

5. 5 Discrete Fourier transform PERIODIC WAVEFUNCTIONDISCRETE SPECTRUM COMPLEX WAVEFUNCTION COMPLEX SPECTRUM

6. 6 Continuous Fourier transform COMPLEX WAVEFUNCTION & SPECTRUM NON-PERIODIC WAVEFUNCTION PERIODIC WAVEFUNCTIONDISCRETE SPECTRUM