Physics 834: Problem Set #8
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.
Recent changes to this page:
- 14-Nov-2011 --- Updates: please read!
- 11-Nov-2011 --- Original version.
- 3-D wave equation.
- This part should be a direct generalization of examples
we've considered or those in Lea in which we take the Fourier
transform of an equation. Don't go to a particular
coordinate system yet (i.e., keep x and k as vectors.
The transform S is given in terms of the transform
F (which is left as a function until parts c and d).
- Assume α > 0. Just find where the poles are!
(What kind of equation do you need to solve?)
- First find F and then set up the inverse Fourier
transform. You might find it simplest to do the ω
integral first (but it is not essential).
This is the same in parts c and d, and uses
the result from part b. Consider t<0 and t>0
separately (one of them is very easy!).
For the k integrals, you'll want to use spherical
coordinates, as in problem 5 from the last problem set.
Use the same trick of choosing the kz axis
so that the angle between k and x is the
polar angle. Do the r integral last and you are
encouraged to use Mathematica (but include a printout if you do).
This is quite similar to the last part, right up to the r
integral. But this one is easy, if you remember about Fourier
transforms and delta functions. Also remember when getting
your answer that t>0.
- Helmhotz equation again.
- You need only consider k2>0 (I
changed my mind). What is the
general solution (with constants)?
If you substitute it into the first boundary condition, you
can eliminate one constant (and the other will multiply everything,
so you can set it equal to one). To be complete, you will
need to consider the possibility separately that a=0.
How does the second boundary conditions determine
the eigenvalue (by a transcendental
equation you don't solve at this point!).
- You should find a closed-form solution for this case.
- If you have a transcendental equation, find the first
few numerical values with Mathematica (try FindRoot).
- Sturm-Liouville derivatives.
- The best way to do this is to show that the differential
equation for the
derivative can be put into the Sturm-Liouville form.
The first step is to find this differential equation.
Note that the original equation can be written in terms
of u and y. A second equation involving u
and y follows by differentiating the original equation.
Then eliminate y.
Now look to put this into the Sturm-Liouville form:
d/dx F du/dx - G u + λ W u = 0, where we've
introduced the new functions F, G, and W.
We can always multiply the equation by a positive
function p(x) to transform a non-self-adjoint operator to
a self-adjoint one. (See section 10.1 of Arfken.)
Determine p(x) by matching the equation for u
with the new form (you will have a differential equation that
you can directly integrate).
- If you've succeeded in the first part, you should
just read off the weighting function W(x).
- If you rewrite the original boundary conditions in
terms of u, you will find that either F must
vanish at a and b or there is a condition
on the α constants (note that you can't have the
eigenvalue appearing in the boundary conditions).
- This last part should be immediate: just identify the
Sturm-Liouville functions from the original
Legendre differential equation to identify the new
weight function W(x) for the derivatives. (It should
agree with the example problem from class!)
- Fourier-Legendre series.
The idea is to relate Il to the integral with
a lower value of l, and then finally evaluating
What is Il for odd l?
The relevant recursion relations are given on pages 377 to 379 of
Lea chapter 8. You will need more than one of them, plus
a partial integration (as with the example from class).
The Fourier-Legendre seres just means an expansion of the
function in terms of Legendre polynomials. So how would you
determined the coefficients of such an expansion (your answer
should use Il).
- Euler-Bernoulli equation.
This one is interesting because you don't often solve differential
equations with fourth derivatives! Otherwise it is conceptually
similar to other problems we've done. You are invited to use
Mathematica here for any integrals.
- Summing a series.
Just follow the instructions. The Fourier transform is a much
simpler sum than the original (think of the p dependence in
the new sum as a power; then it is the Taylor expansion of a
The Mathematica Sum function can do this sum, so be sure
to check your answer against it.
Physics 834: Assignment #8 hints.
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