Physics 834: Problem Set #5

Here are some hints, suggestions, and comments on the assignment. Remember to keep track of the amount of time you spend doing the (entire) assignment and record this number on your problem solution.

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  1. Fourier series for f(x)=x
    1. For an expansion from 0 to L, the equations (4.13), (4.14), and (4.15) in Lea give the expansion coefficients. Mathematica makes quick work of the integrals! But you need to use simplification as in fourier_series.nb to get the coefficients in simplest form. For example:
        f[x_] := x
        an[n_] = FullSimplify[
          2 Integrate[f[x]* Sin[2 Pi n x], {x, 0, 1}],
          Assumptions -> {Element[{n}, Integers]}]
      will give you the an coefficients.
    2. Now you want the range -L < x < L but you need to decide if you want the odd or even expansion, which takes you to equation (4.25) or (4.26)-(4.27). Be careful: those equations already take into account your choice (unlike those in part a) so that they can switch the integral back to the 0 to L interval. (You can also use Lea (4.16) and get the same answer.) If you use Mathematica (recommended!), be careful to use Sin[Pi n x] rather than Sin[2 Pi n x], etc.
    3. The Mathematica notebook fourier_series.nb has examples of such plots. It's efficient to define a function to make your plot if you've defined the coefficients as above (also define b0 and the bn coefficients) :
        fs[x_, nmax_] := b0 + Sum[an[n]*Sin[n*2 Pi x] + bn[n]*Cos[n*2 Pi x], {n, 1, nmax}]
        Plot[{f[x], fs[x, 1], fs[x, 2], fs[x, 3]}, {x, 0, 1}]
      For similarities and differences, observe which expansion does better in different regions (e.g., at the origin), whether they exhibit the Gibbs overshoot, what values they have at a discontinuity, whether there is a constant term or not, and anything else you can think of! Try extending the range of your plots to -2 to 2. This shows you the periodic extension of each function.
  2. Fourier series for f(x)=x2. This is a straightforward application. You might try the complex expansion, but you can also do this just as in problem 1. If you used Mathematica as in the example code above, it's just a matter of changing the function! Example plots in fourier_series.nb.
  3. Complex exponential Fourier series. Again, nothing tricky here if you use the correct formulas for expansion coefficients. Note that you are to use the exponential (complex) expansion intially, then afterward convert to sines and cosines. Mathematica has no trouble with the complex series, just use I for i.
  4. Damped, perioically driven, harmonic oscillator. The first step is to analyze f(t) as a series. Then the coefficients of the x(t) expansion follow algebraically. What type of expansion (exponential or sines and cosines) is best if you have both first and second derivatives? Can you show that your answer for x(t) is real?
  5. Guitar string physics.
    1. You can follow the example in Lea's text, only replace the initial shape of the string. Note that the Fourier series can still be taken as just a sine series because of the boundary conditions at x=0 and x=L.
    2. Are there zero coefficients in the expansion? How about when the string is plucked in the middle? (Think about even and odd harmonics about the center of the string.) Do the coefficients vanish for any values of n? (The Mathematica Table function is useful here to look at a range of n.)
    3. See the fourier_series.nb notebook once again! You can take reasonable values (like unity) for variables like L and v.
  6. Constraints on complex coefficients. We already gave the answer to this in class, but the problem is to carry out the proof. It's straightforward: just take the complex conjugate and use the fact that n is a dummy summation variable.
  7. Using Fourier series to do sums. You don't need to re-derive the series for a step function; just use the result from class or Lea Eq.(4.28). Parseval's theorem for complex Fourier series is given in Eq.(4.32) of the Lea text (but you might find the version in Problem 22 to be more directly usable). If you used a sine/cosine Fourier series, just convert it to a complex series to apply this equation. You can check your answers for the sums using the Sum function in Mathematica. An example of using this function is
    Sum[1/2^m, {m, 0, Infinity}]
    which yields 2, as expected.
  8. Solving the diffusion equation. Follow example 3.15 in Lea to set up the problem and find the general solution. Then it's just a matter of enforcing the boundary condition with the T2 temperature and projecting out (with appropriate integrals!) the coefficients.

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