Announcements 10/18/10 Prayer Found: Physics phor Phynatics book, still unclaimed Term project proposals due on Saturday night!

Email to me: proposal in body of email, 650 word max. See website for guidelines, grading, ideas, and examples of past projects.

Resonator boxes and the Beatles Flame standing wave video from website

http://www.physics.byu.edu/faculty/colton/courses/phy123-fall10/

Colton “Fourier series summary” handout Demo: PVC pipe vs. “Spectrum Lab”

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 Adding cosines together with Mathematica, “sum of

cosines.nb”

http://www.physics.byu.edu/faculty/colton/courses/phy123-fall10/lectures/lecture%2017%20-%20sum%20of%20cosines.nb

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… if someone wants a few extra credit points you could post it to Wikipedia’s group and/or phase velocity articles as an example of group & phase velocities being in opposite directions.

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)

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

What’s a “transform”? A one-to-one correspondence between one function

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

a. If you know one, you can find the other.b. Why? One representation might give you more

insight into the function than the other. 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, and so forth.

c. Why use a Taylor’s series? Sometimes it’s useful.

“Fourier” transform The coefficients of the transform give

information about what frequencies are present

Example: a. my car stereob. my computer’s music player

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