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*To*: AUDITORY@xxxxxxxxxxxxxxx*Subject*: Re: reverse engineering of acoustic sources*From*: Georg Essl <gessl@xxxxxxxxxxxxxxxx>*Date*: Sat, 31 Jan 2004 07:54:04 -0500*Delivery-date*: Sat Jan 31 08:44:03 2004*In-reply-to*: <200401310502.i0V52MRh016917@viroid.mcgill.ca>*References*: <200401310502.i0V52MRh016917@viroid.mcgill.ca>*Reply-to*: Georg Essl <gessl@xxxxxxxxxxxxxxxx>*Sender*: AUDITORY Research in Auditory Perception <AUDITORY@xxxxxxxxxxxxxxx>

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.

**Follow-Ups**:**Re: reverse engineering of acoustic sources***From:*John Neuhoff

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