Fourier series

Fourier series

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Fourier series

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1. Fourier series

2. Fourier Series PDEs Acoustics & Music Optics & diffraction Geophysics Signal processing Statistics Cryptography ...

3. How many of the following are even functions? I: x II: sin(x) III: sin2(x) IV: cos2(x) • None • Exactly one of them • Two of them • Three of them • All four of them!

4. How many of the following are even functions? I: 3x2-2x4 II: -cos(x) III: tan(x) IV: e2x • None • Exactly one of them • Two of them • Three of them • All four of them!

5. What can you predict about the a’s and b’s for this f(t)? f(t) • All terms are non-zero B) The a’s are all zero • C) The b’s are all zero D) a’s are all 0, except a0 • E) More than one of the above (or none, or ???)

6. When you finish P. 3 of the Tutorial, click in: What can you say about the a’s and b’s for this f(t)? f(t) t • All terms are non-zero B) The a’s are all zero • C) The b’s are all zero D) a’s are all 0, except a0 • E) More than one of the above, or, not enough info...

7. What can you say about the a’s and b’s for this f(t)? f(t) t • All terms are non-zero B) The a’s are all zero • C) The b’s are all zero D) a’s are all 0, except a0 • E) More than one of the above!

8. Given an odd (periodic) function f(t),

9. Given an odd (periodic) function f(t), I claim (proof coming!) it’s easy enough to compute all these bn’s:

10. If f(t) is neither even nor odd, it’s still easy:

11. For the curve below (which I assume repeats over and over), what is ω? • 1 • 2 • π • 2π • Something else!

12. Let’s zoom in.Can you guess anything more about the Fourier series?

13. Does this help? (The blue dashed curve is 2Cos πt.)

14. RECAP: Any odd periodic f(t) can be written as: where But why? Where does this formula for bn come from? It’s “Fourier’s trick”!

15. Fourier’s trick: Thinking of functions as a bit like vectors…

16. Vectors, in terms of a set of basis vectors: Inner product, or “dot product”: To find one numerical component of v:

17. Can you see any parallels?

18. Inner product, or “dot product” of vectors: If you had to make an intuitive stab at what might be the analogous inner product of functions, c(t) and d(t), what might you try? (Think about the large n limit?)

19. Inner product, or “dot product” of vectors: If you had to make an intuitive stab at what might be the analogous inner product of functions, c(t) and d(t), what might you try? (Think about the large n limit?) How about: ??

20. What can you say about • 0 B) positive C) negative D) depends • E) I would really need to compute it...

21. If m>1, what can you guess about • always 0 B) sometimes 0 C)???

22. Summary (not proven by previous questions, but easy enough to just do the integral and show this!)

23. Orthogonality of basis vectors: What does ... suggest to you, then?

24. Orthonormality of basis vectors:

25. Vectors, in terms of a set of basis vectors: To find one numerical component: Functions, in terms of basis functions To find one numerical component: (??)

26. Vectors, in terms of a set of basis vectors: To find one numerical component: Fourier’s trick

27. D’oh!

28. To find one component: Fourier’s trick again “Dot” both sides with a “basis vector” of your choice:

29. Given this little “impulse” f(t) (height 1/τ, duration τ), In the limit τ 0, what is 1/τ τ • 0 B) 1 C) ∞ • D) Finite but not necessarily 1 E) ?? Challenge: Sketch f(t) in this limit.

30. What is the value of

31. What is the value of

32. What is the value of

33. What is the value of

34. What is the value of

35. What is the value of

36. Recall that What are the UNITS of (where t is seconds)

37. 1/τ τ

38. PDEs Partial Differential Equations

39. What is the general solution to Y’’(y)-k2Y(y)=0 (where k is some realnonzero constant) • Y(y)=A eky+Be-ky • Y(y)=Ae-kycos(ky-δ) • Y(y)=Acos(ky) • Y(y)=Acos(ky)+Bsin(ky) • None of these or MORE than one!

40. I’m interested in deriving Where does this come from? And what is α? Let’s start by thinking about H(x,t), heat flow at x: H(x,t) = “Joules/sec (of thermal energy) passing to the right through position x” TH TC What does H(x,t) depend on?

41. H(x,t) = Joules/sec (of thermal energy) passing to the right What does H(x,t) depend on? Probably boundary temperatures! But, how? TH TC • H ~ (TH+TC)/2 • H ~ TH - TC (=ΔT) • Both but not in such a simple way! • Neither/???

42. H(x,t) = Joules/sec (of thermal energy) passing to the right What does H(x,t) depend on? Perhaps Δx? But, how? dx TH TC • H ~ ΔT Δx • H ~ ΔT/Δx • Might be more complicated, nonlinear? • I don’t think it should depend on Δx.

43. H(x,t) = Joules/sec (of thermal energy) passing to the right What does H(x,t) depend on? We have concluded (so far) A dx TH TC x Are we done?

44. Heat flow (H = Joules passing by/sec): • How does the prop constant depend on the area , A? • linearly • ~ some other positive power of A • inversely • ~ some negative power of A • It should be independent of area!