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This announcement provides information on term project proposals, including guidelines, grading criteria, and examples. Students are required to submit one proposal per group and CC their partner on the email. More details can be found on the website.
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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? (nothing in particular) Other comments? Is this related to the Heisenberg uncertainty principle? Is the average grade on exam 2 typically higher or lower than exam 1?
Quick Writing • We saw that A1cos(kx + f1) + A2cos(kx + f2) 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?
Special CaseCentered on a particular k: “Wave packets” • HW 21-3 Plot: Explore with Mathematica
Wave packets, cont. • Results: • To localize a wave in space, you need lots of spatial frequencies (k values) • 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"? It goes from infinity to infinity so it's kind of at everywhere at once.
Pure Sine Wave y=sin(5 x) Power Spectrum
“Shuttered” Sine Wave y=sin(5 x)*shutter(x) Power Spectrum Uncertainty in x = ______ Uncertainty in k = ______ (and technically, D = std dev) In general:
Clicker question: • The equation that says DxDk ½ means that if you know the precise location of an electron you cannot know its momentum, and vice versa. • True • False
Uncertainty Relationships • Position & k-vector • Time & w • Quantum Mechanics: momentum p = k energy E = w “” = “h bar” = Plank’s constant /(2p)
Dispersion • A dispersive medium: velocity is different for different frequencies • Any real-world examples? • Why do we care? • Real waves are often not shaped like sine waves. • Non sine-wave shapes are made up of combinations of sine waves at different frequencies. • 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 (w 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: • f(x,t) = cos(x-4t) + cos(2 (x-5t)) • What is “v”? • What is v for w=4? What is v for w=10? • What does that wave look like as time progresses? (next slide)
0.1 seconds 0.7 seconds 1.3 seconds Mathematica What if the two velocities had been the same?
Time Evolution of Dispersive Pulse Note: frequencies are infinitely close together Credit: Dr. Durfee Power spectrum Peak moves at about 13 m/s (on my office computer) How much energy is contained in each frequency component Wave moving in time
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. • 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 Velocity Credit: Dr. Durfee Window is moving along with the peak of the pulse Can be different for each frequency component that makes up the wave 12.5 m/s, for dominant component A property of the wave as a whole (peak) 13 m/s
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. • Example: ex = 1 + x + x2/2! + x3/3! + x4/4! + … • 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 x0 term there is, the second tells you how much of the x1 term there is, etc. • Why Taylor’s series? Sometimes they are useful.
“Fourier” transform • The coefficients of the transform give information about what frequencies are present • Example: • my car stereo • my computer’s music player • your ear (so I’ve been told)
Fourier Transform 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 …” Do the transform (or have a computer do it)
