Announcements 101711 Prayer n Saturday Term project proposals

Announcements 10/17/11 Prayer n Saturday: Term project proposals, one proposal per group… but please CC your partner on the email. See website for guidelines, grading, ideas, examples. n Chris: not here on Friday for office hours n Colton “Fourier series summary” handout. Notation warning! n xkcd

Demos Trumpet, revisited n Gas-lit standing wave n

Reading Quiz n As – – – discussed in the reading assignment, a “beat” is: A periodic change in amplitude of a wave Interference between overtones The first Fourier component of a wave The reflection of a wave from a rigid barrier What the musical “Hairspray” says you can’t stop

Beats n Demo: Tuning forks; Spectrum lab software “beat period” “beat frequency”: fbeat = |f 1 – f 2| (or wbeat = |w 1 – w 2| )

Beats, cont. n Stokes Video (1: 33) http: //stokes. byu. edu/beats_script_flash. html

Beats: Quick Math Can be proved with trig identities carrier “envelope” (beat) Wait… is beat frequency 0. 5 rad/s or is it 1 rad/s? (class poll)

Sine Wave What is its wavelength? What is its frequency? What is its location? When does it occur? Animations courtesy of Dr. Durfee

Beats in Time What is its wavelength? What is its frequency? What is its location? When does it occur?

Localization in Position/Wavenumber What is its wavelength? What is its frequency? What is its location? When does it occur?

Beats in Both. . .

Pure Sine Wave y=sin(5 x) Power Spectrum

“Shuttered” Sine Wave y=sin(5 x)*shutter(x) Uncertainty in x = ______ In general: Power Spectrum Uncertainty in k = ______ (and technically, D = std dev)

Reading Quiz n The equation that says Dx. Dk ½ means that if you know the precise location of an electron you cannot know its momentum, and vice versa. – True – False

Uncertainty Relationships n Position & k-vector n Time & w n Quantum Mechanics: momentum p = k “ ” = “h bar” = Plank’s constant /(2 p) energy E = w

Transforms A one-to-one correspondence between one function and another function (or between a function and a set of numbers). – If you know one, you can find the other. – The two can provide complementary info. n Example: ex = 1 + x 2/2! + x 3/3! + x 4/4! + … – If you know the function (ex), you can find the Taylor’s series coefficients. – If you have the Taylor’s series coefficients (1, 1, 1/2!, 1/3!, 1/4!, …), you can re-create the function. The first number tells you how much of the x 0 term there is, the second tells you how much of the x 1 term there is, etc. – Why Taylor’s series? Sometimes they are useful. n

“Fourier” transform The coefficients of the transform give information about what frequencies are present n Example: – my car stereo – my computer’s music player – your ear (so I’ve been told) n

Fourier Transform Do the transform (or have a computer do it) Answer from computer: “There are several components at different values of k; all are multiples of k=0. 01. k = 0. 01: amplitude = 0 k = 0. 02: amplitude = 0 … … k = 0. 90: amplitude = 1 k = 0. 91: amplitude = 1 k = 0. 92: amplitude = 1 …”