Announcements 2/25/11 Prayer Due Saturday night: a. Labs 4-5 b. First extra credit papers - Can do each type of paper once in first half of semester, once in second half c. Term project proposals – Email to me: proposal in body of email, 650 word max. – One proposal per group… but please CC your partner(s) on email. – See website for guidelines, grading, ideas, and examples of past projects. Exam 2 starts next Saturday! a. Exam 2 optional review session: vote on times by Sunday night; I’ll make a decision Monday morning. Anyone need Colton “Fourier series summary” handout? Spectrum Lab on laptop
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Announcements 2/25/11 Prayer Due Saturday night: a. a.Labs 4-5 b. b.First extra credit papers - Can do each type of paper once in first half of semester,
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Announcements 2/25/11 Prayer Due Saturday night:
a. Labs 4-5b. First extra credit papers - Can do each type of paper
once in first half of semester, once in second halfc. Term project proposals
– Email to me: proposal in body of email, 650 word max.– One proposal per group… but please CC your
partner(s) on email. – See website for guidelines, grading, ideas, and
examples of past projects. Exam 2 starts next Saturday!
a. Exam 2 optional review session: vote on times by Sunday night; I’ll make a decision Monday morning.
Anyone need Colton “Fourier series summary” handout? Spectrum Lab on laptop
Reading Quiz In the Fourier transform of a periodic
function, which frequency components will be present?
a. Just the fundamental frequency, f0 = 1/period
b. f0 and potentially all integer multiples of f0
c. A finite number of discrete frequencies centered on f0
d. An infinite number of frequencies near f0, spaced infinitely close together
e. Masspacity
Fourier Theorem Any function periodic on a distance L can be written
as a sum of sines and cosines like this:
Notation issues: a. a0, an, bn = how “much”
at that frequencya. Time vs distanceb. a0 vs a0/2c. 2/L = k (or k0)… compare 2/T = (or 0 )d. Durfee:
– an and bn reversed– Uses 0 instead of L
The trick: finding the “Fourier coefficients”, an and bn
01 1
2 2( ) cos sinn n
n n
nx nxf x a a b
L L
01
compare to: ( ) nn
n
f x a a x
Applications (a short list) “What are some applications of Fourier transforms?”
a. Electronics: circuit response to non-sinusoidal signals (mentioned last lecture)
b. Data compression (as mentioned in PpP)
c. Acoustics: guitar string vibrations (PpP, next lecture)
d. Acoustics: sound wave propagation through dispersive medium
e. Optics: spreading out of pulsed laser in dispersive medium
f. Optics: frequency components of pulsed laser can excite electrons into otherwise forbidden energy levels
g. Quantum: “particle in a box” situation, aka “infinite square well”--wavefunction of an electron
How to find the coefficients
What does mean?
What does mean?
0
0
1( )
L
a f x dxL
0
2 2( )cos
L
nnx
a f x dxL L
0
2 2( )sin
L
nnx
b f x dxL L
01 1
2 2( ) cos sinn n
n n
nx nxf x a a b
L L
0
0
1( )
L
a f x dxL
1
0
2 2( )cos
Lx
a f x dxL L
Let’s wait a minute for derivation.
Example: square wave
f(x) = 1, from 0 to L/2 f(x) = -1, from L/2 to L
(then repeats) a0 = ? an = ? b1 = ? b2 = ? bn = ?
0
0
1( )
L
a f x dxL
0
2 2( )cos
L
nnx
a f x dxL L
0
2 2( )sin
L
nnx
b f x dxL L
01 1
2 2( ) cos sinn n
n n
nx nxf x a a b
L L
004/Could work out each bn individually, but why?
4/(n), only odd terms
Square wave, cont.
Plots with Mathematica:http://www.physics.byu.edu/faculty/colton/courses/phy123-winter11/lectures/lecture 22 - square wave