Re: reverse engineering of acoustic sources

[Georg Essl <gessl@xxxxxxxxxxxxxxxx>]

Hi Jim,

I cannot speak to the claims about the universe and I have yet to get hold
of this article. But regarding the problem of determining the shape of a
vibrating structure from its spectrum: This has a sizeable literature in
mathematics and many currently active contributers to it.

A very recent but also very technical review is being written by Steve
Zelditch of JHU and available online at:

http://mathnt.mat.jhu.edu/zelditch/Preprints/SurveyJDGAMS4.pdf

a brief and somewhat more readable survey by Ivana Alexandrava can be
found here:

http://math.berkeley.edu/~alanw/240papers00/alexandrova.pdf

Unfortunately I know of very little writing in this field that is less
reliant on the language, background and notation of contemporary pure
mathematics (if anybody knows good "descriptive" surveys, I'd love to hear
about it!)

Since Carolyn Gordon and coworkers gave explicit constructions of "drums
that sound the same" it is known that in general the inverse doesn't hold
in the 2-D case. But the constructions that are used are based on
triangles and hence don't have a smooth boundary. The question of the
concave smooth contour is still unsolved though Zelditch, who wrote the
above survey, recently reported some result in this direction. However
even a shape like a guitar top plate is not yet solved, as Zelditch's
proofs require that what mathematicians call the "length spectrum" (the
set of lengths of closed dynamic trajectories on the geometry) only have
non-degenerate lengths which keeps his results from being much more
general.

It's noteworthy, that even the forward spectral result from geometry is
not generally solved or exhibits symptoms that illustrate the difficulty
of the forward and the inverse problem. E.g the stadium billiard (two
semi-circles connected by straight lines) as representing the dynamics of
a membrane of that shape has chaotic dynamics. I highly recommend the
beautiful paper by Michael Berry on the dynamical result with respect to
chaos of deforming a circle into a stadium:

http://www.phy.bris.ac.uk/research/theory/Berry/the_papers/Berry102.pdf

But to go back to the paragraph at hand, with the caveat of not knowing
the context of the paragraph in their article, at least with respect to
acoustics there is still a lot not known precisely.

- Georg
georg(at)mle.media.mit.edu (current)

> Date:    Fri, 30 Jan 2004 14:24:03 -0600
> From:    beauchamp james w <jwbeauch@UX1.CSO.UIUC.EDU>
> Subject: reverse engineering of acoustic sources
>
> In the Februrary, 2004 issue of Scientific American, Wayne Hu and
> Martin White in their article "The Cosmic Symphony" write
>
>   "...researchers have been able to use [temperature variations of
>   the cosmic microwave background] to precisely estimate the age,
>   composition and geometry of the universe. The process is analogous
>   to determining the construction of a musical instrument by
>   carefully listening to its notes."
>
> My question is: Has there been any work in the second area? (I know
> the reverse has been worked on.) Are the authors trying to say that
> the cosmic quest is as difficult as the acoustic one, or as easy?
>
> Jim Beauchamp
> Univ. of Illinois Urbana-Champaign
>
> PS I note that in their p. 48 figure they have the wrong series of
>    acoustic standing waves.