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Standing Waves

T. T. Stretched string. Wave Equation. Newton’s 2nd. Progressive Waves. Standing Waves. Joseph Fourier, 1768-1830. Fourier Analysis. Normal Modes:. n = 1. n = 3. n = 2. Any wave motion on the string can be described by a sum of these modes!. One equation, infinity unknowns …

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Standing Waves

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  1. T T Stretched string Wave Equation Newton’s 2nd Progressive Waves Standing Waves

  2. Joseph Fourier, 1768-1830 Fourier Analysis Normal Modes: n = 1 n = 3 n = 2 Any wave motion on the string can be described by a sum of these modes! One equation, infinity unknowns … …thanks for nothin’ Joe!

  3. Focus on the shape first! y(x) How to find a specific Bn? Multiply by nth harmonic and integrate. This reduces series to 1 term! Mathematically…

  4. …all terms zero! …uh oh…. 0 x L’Hopital’s Rule: If f(c)=0 and g(c)=0 then

  5. Any shape y(x) between 0 and L

  6. Graphically…. n=6 n=5 n=4 n=3 n=2 n=1 n*=1 Positive contribution zero contribution zero contribution zero contribution zero contribution zero contribution n*=2 zero contribution Positive contribution zero contribution zero contribution zero contribution zero contribution “orthogonal”

  7. 0 L c= slope Integrate by parts:

  8. 1st

  9. 1st + 2nd

  10. 1st + 2nd + 3rd

  11. 1st + 2nd + 3rd + 4th

  12. 1st + 2nd + 3rd + 4th + 5th

  13. Barry’s profile

  14. Barry’s profile

  15. Barry’s 1st harmonic

  16. Barry’s 1st and 2nd harmonic

  17. Barry’s 1st, 2nd, and 3rd harmonic

  18. clear load f; f=f-172; plot(f) B(200)=0; y(355)=0; for n = 1:200 for x = 1:355 B(n)=B(n)+(2/355)*f(x)*sin(n*pi*x/355); end end figure plot(B,'.') for x = 1:355 for n = 1:200 y(x)=y(x)+B(n)*sin(n*pi*x/355); end end figure plot(y)

  19. Les Foibles de Fourier en Frances (French’s Fourier Foibles) Good choice if boundaries are at 0: Not so good for other shapes..

  20. More General: function periodic on interval x = –L to x = L.

  21. H x 0 L/8 L Something hard:

  22. MATLAB summation of 1 to 100 terms. H 0 0 L

  23. Any well behaved repetitive function can be described as an infinite sum of sinusoids with variable amplitudes (a Fourier Series). On a stretched string these correspond to the normal modes. Fourier analysis can describe arbitrary string shapes as well as progressive waves and pulses.

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