Physics 8 03 Vibrations and Waves Lecture 11 Fourier Analysis with traveling waves Dispersion Last time Arbitrary motion Superposition of ALL possible normal modes m 1 n 0 y x t Am sin k m x cos mt m Bn cos k n x cos nt n Orthogonal functions Fourier coefficients L 2 Am L y x t 0 sin k x dx 2 Bn L L m 0 y x t 0 cos k x dx n 0 Fourier expansion recip Start with superposition of all possible modes Determine the simplest basis functions using Boundary conditions 0 L or L 2 L 2 or L L Symmetry f x 0 f x 0 or f x 0 f x 0 Initial condition y x 0 0 or vy x 0 0 Determine the Fourier coefficients An and or Bn Use orthogonality relations with Initial deformation y x t 0 or Initial velocity vy x t 0 Add the time dependence Fourier expansions for traveling waves What happens if the Fourier components all travel at slightly different speeds n v kn DISPERSION Wave equation in dispersive media Phase velocity velocity of a single crest of the wave with average wave vector k vp k Group velocity velocity of the slow envelope velocity of energy transport d vg dk Corrections comments on today s lecture Formula for approximation of m was written incorrectly on the board the correct version is 2 c 2 k m2 1 k m2 ck m 1 k m2 ck m 1 1 k m2 2 Where does the equation for a stiff string come from For a derivation see for example Fetter and Walecka Theoretical mechanics of Particles and Continua page 221
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