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

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

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

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

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

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

7. 0 L Integrate by parts: c = slope

9. 1 st

10. 1 st + 2 nd

11. 1 st + 2 nd + 3 rd

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

13. 1 st + 2 nd + 3 rd + 4 th + 5 th

15. Barry’s profile

17. Barry’s 1 st harmonic

18. Barry’s 1 st and 2 nd harmonic

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

21. 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 figure plot(B,'.') for x = 1:355 for n = 1:200 y(x)=y(x)+B(n)*sin(n*pi*x/355); end figure plot(y)

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

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

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

26. 0 L 0 H MATLAB summation of 1 to 100 terms.

27. 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.