PROBLEM SET 5 COMMENTS - SPRING 2003

	There are many possible sources for instructions on how to
	find Fourier coefficients.  If you are taking or have taken
	any version of 18.03, that text may have much that is useful
	as well.  For instance, if you have the fourth edition of the
	Edwards & Penney text, the endpapers have useful integrals.

	If you consult the B&B text (which you really should, at some
	point), beware of Figure 2.22(a) on Page 172;  the positive
	direction is down in the figure!  This is certainly different,
	but it doesn't affect the math or physics one bit.

	5.1:  First off, make sure you're looking at a version of the
          problem set with the result of part (a) corrected.
	  For this problem in general, the idea is that the number of
	  modes is approximated by a continous function of the
	  wavenumber k.  In two or three dimensions (parts (b) and
	  (c)), the term "k" should be taken as the magnitude of a
	  vector.
	  The source of this problem is French Problem 6-13, but some
	  printings of the text have the result terribly garbled
	  (confusion between the frequency "nu" and the speed
	  "vee").  See the notes, linked from the 8.03-ESG main page,
	  for a corrected version.
	  For a generalization that is really quite valid, use of "k"
          instead of "nu" or "omega" is preferable;  the problem as
          given has a much cleaner result than the similar result
          using other parameters.
	  The corresponding discussion in B&B is on Pages 305-306,
          leading up to Equation (4.65).  This discussion is a bit
          more involved, in that it considered EM waves (v=c), not
          mechanical waves, and assumes a more general rectangular
          parallelepiped, as opposed to a cube.  Also, the result is
          in terms of omega instead of k, but from v=omega/k the
          desired result follows.  In going from one notation to the
          other, what the Problem 5.1 calls "rho" would be dN/dk.

	4.2:  Everyone has to do such Fourier Series at least once,
          after which you should appreciate why some of us prefer to
          have a machine to the work.
	  There's a slightly subtle point involved, one not usually
          covered in typical physics classes.  The fact is, when a
          function is defined on a finite interval, there are many
          possible Fourier Series representations, depending on how
          the fuction is extended periodically.  For instance, in part
          (c), a Fourier Series on the interval 0 to L/2 is trivial,
          but getting the series to vanish on the interval L/2 to L
          requires more terms (in fact an infinite number of terms).
	  For our purposes, assume that the given functions are
          extended to periodic functions that are odd about x=0 and
          x=L;  that is, don't do anything too complicated.  Use
          functions of the form sin(n*x*Pi/L) for your basis
          functions;  this makes life easier, or at least less
          difficult.

	4.3:  The fact is, if you were paying attention and taking
          notes in lecture March 10, or if you know where to look in
          B&B, this one is done for you.

	4.4 really is straightforward, with some math and a great deal
          of discussion.  As indicated, it's a bit ahead of the
          lectures, so you may have to read about "phase velocity" on
          your own.  You have at least two sources for this.