Warm-Up Exercise 22

Due 12:15 pm, Wed 17 Oct 2012

Physics 123, Fall 2012

Reading assignment: PpP 6.1-6.5

When you Fourier transform a function, the result is a sum of sines and cosines. The picture above shows one possible result. For this case, the sum is of cosines only, and the amplitudes (the "coefficients") of the cosine terms mainly decrease from one term to the next. The decrease is given by the "Sinc[k]^2" term multiplying the Cos[kx] term. ("Sinc[k]" means "(1/k) Sin[k]", in case you haven't seen that before.)  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 sum does yield the periodic "tent" function shown. 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. (You can do that by specifying a step size in the Sum command, such as {k,1,50,2}... that will only sum the odd terms.) Write down the formula for the most interesting function that you created.

I thought this next function was pretty interesting. Looks a bit like Batman! :-) f[x_] = Sum[Sinc[k] Cos[k x], {k, 1, 200, 4}];

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 top plot is the actual voltage vs. time signal recorded by the microphone. The bottom plot is the Fourier transform, which shows all of the *amplitudes* of the various frequency components that would have to be added together to form that signal.

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