Announcements 2/23/11 Prayer Deron Williams just got traded to the Nets (what???) 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.

Thanks to those who filled out the mid-semester survey Colton “Fourier series summary” handout. Caution

about notation!!

Demos Trumpet 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. Video:

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

Review: Wave packets HW 17-5

Review of wave packets, cont.

What did we learn?a. To localize a wave in space, you need lots of

frequenciesb. To remove neighboring localized waves, you

need those frequencies to spaced close to each other. (infinitely close, really)

Review: How did I create this?

Cos1.23457 t 0.9 x Cos1.20758 t 0.91 x Cos1.18147 t 0.92 x Cos1.1562 t 0.93 x

Cos1.13173 t 0.94 x Cos1.10803 t 0.95 x Cos1.08507 t 0.96 x

Cos1.06281 t 0.97 x Cos1.04123 t 0.98 x Cos1.0203 t 0.99 x Cos1. t 1. x

Cos0.980296 t 1.01 x Cos0.961169 t 1.02 x Cos0.942596 t 1.03 x

Cos0.924556 t 1.04 x Cos0.907029 t 1.05 x Cos0.889996 t 1.06 x

Cos0.873439 t 1.07 x Cos0.857339 t 1.08 x Cos0.84168 t 1.09 x Cos0.826446 t 1.1 x

1500 1000 500 500 1000 1500

20

10

10

20

What I didn’t show you:(zoomed out)

Still mesmerizing… and extra credit still up for grabs

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 “Uncertainty Principle” from quantum mechanics (if you know exactly where a particle is, you can’t know exactly what its momentum is, and vice versa) is a result of the xk equation just discussed.

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 “transform” is: 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…”

600 400 200 200 400 600

20

10

10

20

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