Warm-Up Exercise 22

Due 10:00 am, Fri, Feb. 24

Physics 123, Winter Semester, 2012

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When you Fourier transform a function, the result is a sum over sines and cosines. The picture above shows one possible result---the coefficients for the sine terms is zero and the terms for the cosines decay as the square of the sinc function. Note that the argument for the coefficients is the wave vector k. Type the above code into Mathematica (use Remote Desktop Connection if it's not convenient to go to the computer lab). Verify that this particular cosine sum yields a periodic "tent" function. Play around with other forms for the coefficients. For example, try things like 1/k^2, 1/k, exp(-k). Try using only odd (or even) values of k. Play around and then describe your results here. (Can you choose the coefficients so that they make square pulse? Like Fig. 6.3?)

Take a look at Fig. 6.2 again. Do you understand what the two plots are showing? If so, explain. If not, ask a question here.

The math was kind of thick and heavy in 6.4. Read it again and keep the following in mind---there are really only two main steps: 1) multiply by another cosine (but of a different frequency), and 2) integrate the product over a period. (When I was a student, my friend York called this process "multigrating", a nice mnemonic.) And the important thing to realize is that the product of two cosines, of different frequecies, integrated over an integral number of periods, will give you zero. So you can pick off the terms one by one just by changing the frequency of the cosine you used in step one. If you still have questions after re-reading, ask them here.

Was there anything that you didn't understand in the reading assignment? If so, explain.

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