Announcements 10/15/12 Prayer Saturday: Term project proposals, one proposal per group… but please CC your partner on the email. See website for guidelines,

Announcements 10/15/12 Prayer Saturday: Term project proposals, one proposal per

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

Colton “Fourier series summary” handout. Notation warning!

xkcd

From warmup

Extra time on?a.(nothing in particular)

Other comments?a.Is this related to the Heisenberg

uncertainty principle?b.Is the average grade on exam 2 typically

higher or lower than exam 1?

Spectrum Lab Software

Quick Writing We saw that

A1cos(kx + 1) + A2cos(kx + 2) gives you a cosine wave with the same k, and hence wavelength.

If you add a third, fourth, fifth, etc., such cosine wave, you still get a simple cosine wave. See

How can you then add together cosine waves to get a more complicated shape with same wavelength? Or can you?

If not all multiples of same k:

Special Case Centered on a particular k: “Wave packets”

HW 21-3

Plot:

Explore with Mathematica

Wave packets, cont.

Results:a. To localize a wave in space, you need lots of spatial

frequencies (k values)b. To remove neighboring localized waves (i.e. to

make it non-periodic), you need those frequencies to spaced close to each other. (infinitely close, really)

From warmup

PpP states that "a pure sine wave has a precisely defined frequency... but a completely undefined position." What does it mean to have a "completely undefined position"?

a. It goes from infinity to infinity so it's kind of at everywhere at once.

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

Clicker question:

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

Dispersion A dispersive medium: velocity is different for

different frequenciesa. Any real-world examples?

Why do we care? a. Real waves are often not shaped like sine

waves.– Non sine-wave shapes are made up of

combinations of sine waves at different frequencies.

b. Real waves are not infinite in space or in time.– Finite waves are also made up of combinations

of sine waves at different frequencies.Focus on (b) for now… (a) is the main topic of the “Fourier transform” lectures

Dispersion Review Any wave that isn’t 100% sinusoidal contains

more than one frequencies. To localize a wave in space or time, you need

lots of frequencies--spatial (k values) or angular ( values), respectively. Really an infinite number of frequencies spaced infinitely closely together.

A dispersive medium: velocity is different for different frequencies.

Two Different Velocities What happens if a wave pulse is sent through a

dispersive medium? Nondispersive? Dispersive wave example:

a. f(x,t) = cos(x-4t) + cos(2 (x-5t)) – What is “v”?– What is v for =4? What is v for =10?

What does that wave look like as time progresses? (next slide)

Mathematica

0.7 seconds 1.3 seconds

0.1 seconds What if the two velocities had been the same?

Time Evolution of Time Evolution of DispersiveDispersive Pulse PulseCredit: Dr. Durfee

Wave moving in time

Peak moves at about 13 m/s (on my office computer)

How much energy is contained in each frequency component

Power spectrumNote: frequencies are infinitely close together

From warmup: phase & group velocities

Examples where vphase vgroup: http://en.wikipedia.org/wiki/Group_velocity

For each figure, measure the speed of travel on your monitor (cm/s) for both the envelope and the ripples.

a. Your results will depend on size of monitor and/or zoom level. But the ratio of envelope to ripple speed should be the same as me.

– Top Fig: speed of envelope (green dots) = 0.46 cm/s, speed of ripples (red dot) = 2x that

– Second Fig: speed of envelope = 1.50 cm/s; speed of ripples = -0.33x that

Phase and Group VelocityPhase and Group Velocity

evaluated at

velocity of "envelope"ave

gk

dv

dk

velocity of "wiggles"pv k

Credit: Dr. Durfee

Can be different for each frequency component that makes up the wave

A property of the wave as a whole

Window is moving along with the peak of the pulse

13 m/s

12.5 m/s, for dominant component

(peak)

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

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