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## Fourier series

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**Fourier Series**PDEs Acoustics & Music Optics & diffraction Geophysics Signal processing Statistics Cryptography ...**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!**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!**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 ???)**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...**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!**Given an odd (periodic) function f(t),**I claim (proof coming!) it’s easy enough to compute all these bn’s:**For the curve below (which I assume repeats over and over),**what is ω? • 1 • 2 • π • 2π • Something else!**Let’s zoom in.Can you guess anything more about the**Fourier series?**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”!**Fourier’s trick:**Thinking of functions as a bit like vectors…**Vectors, in terms of a set of basis vectors:**Inner product, or “dot product”: To find one numerical component of v:**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?)**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: ??**What can you say about**• 0 B) positive C) negative D) depends • E) I would really need to compute it...**If m>1, what can you guess about**• always 0 B) sometimes 0 C)???**Summary (not proven by previous questions, but easy enough**to just do the integral and show this!)**Orthogonality of basis vectors:**What does ... suggest to you, then?**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: (??)**Vectors, in terms of a set of basis vectors:**To find one numerical component: Fourier’s trick**To find one component: Fourier’s trick again**“Dot” both sides with a “basis vector” of your choice:**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.**Recall that**What are the UNITS of (where t is seconds)**1/τ**τ**PDEs**Partial Differential Equations**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!**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?**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/???**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.**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?**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!