Wave Physics PHYS 2023 Tim Freegarde

Wave Physics PHYS 2023 Tim Freegarde

Wave Physics WAVE EQUATIONS & SINUSOIDAL SOLUTIONS WAVE PROPAGATION BEHAVIOUR AT INTERFACES SUPERPOSITIONS FURTHER TOPICS general wave phenomena wave equations, derivations and solution sinusoidal wave motions complex wave functions Huygens model of wave propagation interference Fraunhofer diffraction longitudinal waves continuity conditions boundary conditions linearity and superpositions Fourier series and transforms waves in three dimensions waves from moving sources operators for waves and oscillations further phenomena and implications http://www.avcanada.ca/albums/displayimage.php?album=topn&cat=3&pos=7 2

Spectrum periodic waveform can be expressed as a superposition of harmonics commonly represented by spectrum often plotted on log-log or lin-log scales may be plotted with frequency axis upwards...... with marks for integer...... and time on the horizontal axis 3

Stopped organ pipe even harmonics are absent CLARINET: register key raises by a twelfth 4

Oboes and saxophones even harmonics are present CLARINET: register key raises by a twelfth SAXOPHONE: register key raises by an octave 5

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

Fourier requirements LINEARITY of wave equation if and are solutions, then so is ORTHOGONALITY of solutions if COMPLETENESS of solutions any function may be written as a superposition common basis sets: sinusoidal complex exponential FOURIER Laguerre Hermite 7

Fourier synthesis any function may be written as a superposition 8

Fourier analysis any function may be written as a superposition coefficients or etc. are found using the property of orthogonality multiply waveform by function integrate product over range of and normalize whose coefficient is required use, or as appropriate complications...: periodic and non-periodic waveforms complex exponential basis functions limits or factors of 9

Fourier transform variations* terminology: transform analysis synthesis function periodicity: discrete transform continuous transform integration limits: scale factors: reference functions: signs: * no correlations between rows ie transform doesn t mean scale factor = 1 etc. 10

Fourier transform Square wave a -T/2 T/2 t -a 11

Amplitude and power spectra amplitude spectrum power/intensity spectrum Square wave 12

Discrete Fourier transform PERIODIC WAVEFUNCTION DISCRETE SPECTRUM COMPLEX SPECTRUM COMPLEX WAVEFUNCTION 13

Continuous Fourier transform PERIODIC WAVEFUNCTION DISCRETE SPECTRUM NON-PERIODIC WAVEFUNCTION COMPLEX WAVEFUNCTION & SPECTRUM 14

Beating TWO DIFFERENT FREQUENCIES ω1 ω2 ω1 + ω2 cosω1t + cosω2t = cos t cos 2 2 t 15

Bandwidth theorem 2 1.5 ν = 0.10 Hz 1 0.5 0-40 -30-20 -10 0 10 20 30 time 40-0.5-1 -1.5 amplitude 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 frequency -2 6 4 2 0-40 -30-20 -10 0 10 20 30 time 40-2 amplitude 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0-4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 frequency -6 16

Bandwidth theorem 2 1.5 ν = 0.20 Hz 1 0.5 0-40 -30-20 -10 0 10 20 30 time 40-0.5-1 -1.5 amplitude 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 frequency -2 6 4 1 0.9 2 0-40 -30-20 -10 0 10 20 30 time 40-2 amplitude 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0-4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 frequency -6 17

Bandwidth theorem 2 1.5 ν = 0.05 Hz 1 0.5 0-40 -30-20 -10 0 10 20 30 time 40-0.5-1 -1.5-2 amplitude 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 frequency 6 4 2 0-40 -30-20 -10 0 10 20 30 time 40-2 amplitude 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0-4 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 frequency -6 18

Bandwidth theorem uncertainty = standard deviation in = root mean square deviation bandwidth theorem: for conjugate variables, (time, frequency): (position, wavenumber): equality holds for Gaussian pulses 19

Bandwidth theorem uncertainty = standard deviation in = root mean square deviation 2015 results! bandwidth theorem: for conjugate variables, (time, frequency): (position, wavenumber): equality holds for Gaussian pulses 20

Bandwidth theorem uncertainty = standard deviation in = root mean square deviation last year s results! bandwidth theorem: for conjugate variables, (time, frequency): (position, wavenumber): equality holds for Gaussian pulses 21

Bandwidth theorem uncertainty = standard deviation in = root mean square deviation this year s results! bandwidth theorem: for conjugate variables, (time, frequency): (position, wavenumber): equality holds for Gaussian pulses 22

Thermal waves copper bar

Fourier transforms Any waveform may be expressed as... a function of time [position]......or... a function of frequency [wavenumber]...where most physical examples are the real part For a single or isolated frequency component, Dirac δ-function 24

Convolution 25

Convolution 26

Convolution theorem Convolution theorem 27

Diffraction grating y ϑ x d λ δs place secondary sources along wavefront...and trigger when wavefront arrives apply to sinusoidal waves by taking into account the phase with which components arrive combine by adding the amplitudes contributions may therefore interfere constructively or destructively 28

Diffraction gratings 29

Diffraction gratings m=-1 1 0.8 0.6 0.4 m=0 m=1 m=-2 0.2 m=2 ϑ -1.5-1 -0.5 0.5 1 1.5 newport.com 30

2-D Fourier transforms 4096 pixels 4096 components add 2-D sinusoidal components in order of strength superposition component

2-D Fourier transforms 4096 pixels 4096 components add 2-D sinusoidal components in order of strength superposition component

2-D Fourier transforms 4096 pixels 4096 components add 2-D sinusoidal components in order of strength superposition Fourier transform optical information processing Photonics Labs image compression e.g. JPEG computer tomography (CT)

2-D Fourier transforms discovery of structure of DNA R E Franklin & R G Gosling, Nature 171, 740 (1953) superposition diffraction pattern optical information processing Photonics Labs image compression e.g. JPEG computer tomography (CT) x-ray diffraction