Re: [Phys-L] Laplace equation problem

[John Denker <jsd@av8n.com>]

My father's favorite joke:

  Q:  What's green and has six wheels?

  A:  Grass.  I lied about the wheels.

He had about 100 more jokes in the same category.  He thought they were
hysterically funny. 

Also:
  Q:  How do you calculate the capacitance of a cow?

  A:  It's easy.  Start by assuming a spherical cow.......

I mention this because on 04/02/2013 04:24 PM, Carl Mungan wrote:
> I am planning to put the following problem on a test tomorrow and I'm
> somewhat surprised by the answer. So let me ask the list for insight:
>
> A square waveguide extends from -infinity to infinity in the z direction.
> It has two grounded plates at y=-0.5 and at y=0.5. The plate at x=-0.5 has
> potential -sin(pi*y). The plate at x=0.5 has potential +sin(pi*y). What is
> the surface charge density on the plate at y=-0.5?
>
> (Actually the first part of the problem asks one to find the potential
> everywhere inside the waveguide. I give them the hint to just write down
> the appropriate form to match the boundary conditions. For those a bit
> rusty on this kind of problem, this means a sinh in the x direction and a
> sine in the y direction, with each coordinate multiplied by pi. You can now
> find the overall scaling constant by fitting to the boundaries.)

This strikes me as seriously ill-specified.  It's something of an ESP exam,
because it hinges on reading the instructor's mind, to find out what was
intended.

There may be a cultural issue here.  It is important to know the rules.  As
the saying goes, if you're supposed to be playing football, don't show up
with your baseball bat and glove.

I assume today's game is supposed to be played by ivory tower rules.  That 
is, we are supposed to fill in all the missing information by making the 
simplest and most favorable assumptions.

  Tangential remark:  I hope that at some point, before these students go 
  out and take command of nuclear submarines, they learn to play by different 
  rules.

Returning now to the spherical cow in the ivory tower:

 1) I assume the word "waveguide" means one thing and the word
  "plate" means something else.

 2) I assume the boundary conditions are independent of z.

 3) I assume the desired solution is independent of z.

 4) I assume the size of the waveguide is 1 unit in whatever units
  we are using.

 5) As a consequence of (2) and (4), I assume the "plates" are
  coextensive with the walls of the waveguide.   This contradicts (1).

 6) I assume the boundary conditions are independent of t.  This is
  sort of implied by the word "Laplace" in the subject line.

 7) I assume all the plates are perfectly conducting.

 8) I assume the boundary voltage is as stated.

Now we have a problem, because the specified voltage pattern is inconsistent
with the assumptions.  The voltage distribution is about what you would expect 
for a traveling wave in a waveguide ... but the Subject: line suggests DC.  So 
where's the gag?  Is it not Laplace?  Is it not DC?  Is it not a waveguide?  Is 
it not green?  Does it have less than 6 wheels?

I'm not trying to be nitpicky here.  The first time I read the question, I 
thought the waveguide was much larger than the plates, i.e. the plates were
just little antennas sticking into the waveguide to launch the wave.  I assumed
that the switch from "waveguide" terminology to "plate" terminology must mean
something.  It was only after seeing the intended solution that I was able to 
work backward and figure out what the question must have been.

I predict that students are going to waste huge amounts of time trying to
decode the statement of the problem.  Everybody likes "rich context" problems
when the context is real enough that you can use real-world intuition.  OTOH
when real-world terminology such as "waveguide" is painted onto a completely
fake problem, the more you know about the real world the less able you are to 
solve the problem.

My suggestions:
 -- If the problem is z-invariant, say so.
 -- If the problem is t-invariant, say so.
 -- If the boundary conditions are incompatible with real-world waveguides,
  don't call it a waveguide.
 -- If the boundary charge distribution is arbitrary and artificial, say so.
 -- If the plates are coextensive with the walls, pick one set of terminology
  and stick with it:  either 100% plates or 100% walls.  Failing that, at
  least define the terms clearly enough that we can see that they are
  interchangeable.
 -- If *one* pair of walls is perfectly conducting and the other is not,
   say so.

=====================

If the original question meant to ask for intuition about the sinh .....

1) My response to that is, what else could it be?  By symmetry it has to be an
odd function.  How many odd functions are there that are the solution to a
homogeneous second-order linear differential equation?  The first things I
would guess are 
  a) straight line
  b) sin
  c) sinh

2) In more scientific terms:

 -- This is a 2D problem.  The z dimension is just there to distract the tourists.

 -- The 2D Laplacian says that the curvature in the x-direction has to just
  exactly cancel the curvature in the y-direction.  Every point is a saddle
  point.  This rules out (1a) and (1b) above.

3) Perhaps even more fundamentally:  The crucial "break" in the process of
solution is guessing that there might be a separation of variables such that
the solution can be written as a product of an x-function and a y-function.

There is no law that says such a factorization is possible in general.  By 
way of counterexample, the atomic p-orbital is certainly not factorable
into an x-function and a y-function.  To say the same thing the other way:
the pseudo-waveguide situation was artfully contrived to *make* factorization 
possible.

So in some sense, the main thing the question is testing is simply this:
 a) Can the student decode the question?
 b) Will the student be clever enough to try a product-form Ansatz?  If yes,
  the rest of the work is trivial.  If not, things are going to get ugly.