Announcements 10/17/11 Prayer Saturday: Term project proposals, one proposal per

group… but please CC your partner on the email. See website for guidelines, grading, ideas, examples.

Chris: not here on Friday for office hours Colton “Fourier series summary” handout. Notation

warning!

xkcd

Demos Trumpet, revisited Gas-lit standing wave

Reading Quiz As discussed in the reading assignment, a

“beat” is:a. A periodic change in amplitude of a waveb. Interference between overtonesc. The first Fourier component of a waved. The reflection of a wave from a rigid

barriere. What the musical “Hairspray” says you

can’t stop

Beats Demo: Tuning forks; Spectrum lab software

“beat frequency”: fbeat = |f1 – f2|“beat period”

(or beat = |1 – 2| )

Beats, cont. Stokes Video (1:33)

http://stokes.byu.edu/beats_script_flash.html

Beats: Quick Math

cos cos 2cos cos2 2

a b a ba b

cos(30 ) cos(31 ) 2cos 30.5 cos 0.5t t t t

carrier “envelope” (beat)

Wait… is beat frequency 0.5 rad/s or is it 1 rad/s? (class poll)

Can be proved with trig identities

Sine WaveSine Wave

What is its wavelength?

What is its location?

What is its frequency?

When does it occur?

Animations courtesy of Dr. Durfee

Beats in TimeBeats in Time

What is its wavelength?

What is its location?

What is its frequency?

When does it occur?

Localization in Position/WavenumberLocalization in Position/Wavenumber

What is its wavelength?

What is its location?

What is its frequency?

When does it occur?

Beats in Both...Beats in Both...

Pure Sine WavePure Sine Wave

y=sin(5 x) Power Spectrum

““Shuttered” Sine WaveShuttered” Sine Wave

y=sin(5 x)*shutter(x) Power Spectrum

Uncertainty in x = ______ Uncertainty in k = ______

1

2x k In general: (and technically,

= std dev)

The equation that says xk ½ means that if you know the precise location of an electron you cannot know its momentum, and vice versa.

a. Trueb. False

Reading Quiz

Uncertainty Relationships Position & k-vector

Time &

Quantum Mechanics: momentum p = k

energy E =

1

2x k

1

2t

“” = “h bar” = Plank’s constant /(2)

2x p

2E t

Transforms A one-to-one correspondence between one

function and another function (or between a function and a set of numbers).

a. If you know one, you can find the other.b. The two can provide complementary info.

Example: ex = 1 + x + x2/2! + x3/3! + x4/4! + …a. If you know the function (ex), you can find the

Taylor’s series coefficients.b. 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 x0 term there is, the second tells you how much of the x1 term there is, etc.

c. Why Taylor’s series? Sometimes they are useful.

“Fourier” transform The coefficients of the transform give

information about what frequencies are present

Example: a. my car stereob. my computer’s music playerc. your ear (so I’ve been told)

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 = 0k = 0.02: amplitude = 0……k = 0.90: amplitude = 1k = 0.91: amplitude = 1k = 0.92: amplitude = 1…”

Cos0.9 x Cos0.91 x Cos0.92 x

Cos0.93 x Cos0.94 x Cos0.95 x

Cos0.96 x Cos0.97 x Cos0.98 x

Cos0.99 x Cos1. x Cos1.01 x Cos1.02 x

Cos1.03 x Cos1.04 x Cos1.05 x Cos1.06 x

Cos1.07 x Cos1.08 x Cos1.09 x Cos1.1 x

600 400 200 200 400 600

20

10

10

20