# 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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