Re: Exotic Drums and the Helmholtz Equation
- From: Igor Khavkine <igor.kh@xxxxxxxxx>
- Date: Wed, 15 Nov 2006 23:46:23 +0000 (UTC)
Greg Egan wrote:
Suppose we have a drum whose membrane is a connected subset M of R^2
with a piecewise smooth boundary, and we assume that the displacement
of the membrane f(x,y,t) obeys a lossless, dispersionless wave
equation:
lap f = (1/v^2) @^2 f / @t^2
where "lap" is the 2-dimensional Laplacian. The boundary condition
is that f is zero on the boundary of M.
If the drum has any pure tones, there will be solutions for f that
separate into the form:
f(x,y,t) = F(x,y) cos(omega t)
and F will be a solution of the Helmholtz equation:
lap F = -k^2 F
where k = omega / v. F will be zero on the boundary of M.
Now, I know how to find analytical solutions of the Helmholtz
equation when the boundary is "nice" -- when it consists of curves of
constant coordinate values, for some coordinate system in which the
equation is separable. So rectangles, circles, annuli, and wedges
are all easy to find solutions for.
In those nice cases, there is a very simple pattern to the modes:
for each coordinate, there can be any positive integer number of
peaks in the standing wave across each coordinate direction. So a
rectangular membrane has modes consisting of n peaks in one direction
and m peaks in the other, with the lowest-frequency mode containing
just a single peak, and F(x,y) is zero only on the boundary.
My two questions are:
(1) Is there an existence theorem which says that there *must* be
solutions of the Helmholtz equation for an arbitarily-shaped
membrane? In other words, are there always eigenfunctions of the
Laplacian with real, negative eigenvalues which are zero on any given
piecewise smooth simple closed curve? Or is it possible to build a
drum which (even under our idealised assumptions) is physically
incapable of producing a pure tone?
I think that under very general conditions on the boundary (including
it being a piecewise smooth simple closed curve) the drum will have
infinitely many pure tones of successively higher frequencies. I can't
give the proof off hand, but I can give a sketch.
Let D be the drum surface. The solution of the Helmholtz equation with
the conditions that you specified comes down to the problem of
diagonalizing the Laplacian operator on the space of square integrable
functions on D, L^2(D), with some caveats. The caveats come from the
fact that the Laplacian is an unbounded operator (recall that taking
derivatives usually makes functions more singular). Since it is not
defined for all elements of L^2(D), we have to choose a dense subspace
on which it is defined. The twice differentiable functions which vanish
at the boundary will do nicely. Note that this choice is affected by
the desired boundary conditions.
To be rigorous, one would now have to show that the Laplacian defined
on this domain can be extended to an unbounded but closed self-adjoint
operator on L^2(D). This part is very technical, but this is indeed
possible for the Laplacian and many other elliptic differential
operators. Once we have the above fact, the spectral theorem can be
applied. Unfortunately, this does not preclude a continuous spectrum.
But even in this case, there will be vibrational solutions which
approximate pure modes to any degree of accuracy.
The last step of the proof would be to convert the
Helmholz equation to an integral one. In this way, we look at the
spectrum not of the Laplacian, but it's inverse, which is a bounded,
even compact, operator on L^2(D). Compactness is a technical property
that guarantees that the spectrum is completely discrete and has zero
as the point of accumulation. A discrete spectrum guarantees the
existence of a complete set of eigenvectors, both of the Laplacian and
its inverse.
The inverse of the Laplacian is the integral operator whose kernel is
the Green function of the Poisson equation with the same boundary
conditions. Its compactness as an operator on L^2(D) follows from
the Green function's regularity (smooth, bounded everywhere except a
neighborhood around the point disturbance, divergence at that point no
stronger, for 2D, than logarithmic). These regularity properties can be
observed by placing a small massive bead on the surface of a drum and
observing its deviation from the unstretched configuration. It seems
reasonable that these regularity properties will be observed for any
physically realizable drum. I think the last conclusion is enough to
answer your original question as I did in my first paragraph above.
(2) If/when solutions exists, must there always be one solution which
is zero only on the boundary (as is certainly the case for "nice"
boundaries)?
I think the answer to this question is also affirmative. One way to see
it is to invoke the heat diffusion equation in place of the wave
equation. Imagine the a conducting plate of the same shape as the drum,
whose boundary is kept at some reference temperature (which can be
taken as zero):
Lap f = @f/@t, f = 0 on the boundary.
The Helmholtz equation is still relevant here because because Laplacian
eigenmodes, Lap f = -k^2 f, will correspond to temperature
distributions which will be uniformly damped by diffusion,
f(t) = exp(-k^2 t) f. Take the eigenmode f with the lowest allowed
value of k, this is the fundamental mode. Because of exponential
damping, for any initial temperature distribution, if we wait long
enough, only the fundamental mode will survive (up to negligible
contributions from other modes which will have decayed much more
quickly). So, if we start with a uniform positive temperature
distribution, as we wait, some heat will leak through the boundary and
and the remnants try to distribute themselves through diffusion all
around the plate. However, the temperature will remain positive
everywhere except the boundary. And, if we wait long enough, the
profile of the temperature distribution will be proportional (up to
exponentially suppressed contributions) to the fundamental mode. From
which it follows that the fundamental mode is everywhere positive (and
hence non-zero), except at the boundary. I believe this property of the
heat diffusion equation is referred as the maximum principle.
Hope this helps.
Igor
.
- References:
- Exotic Drums and the Helmholtz Equation
- From: Greg Egan
- Exotic Drums and the Helmholtz Equation
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