Hi Jim,
I think it's probably fair to say that the pure mathematicians who work
on this don't necessarily have typical real-world acoustical situations in
mind. The formalisms tend to isolate one problem and tend to try to make a
dent there. So 2-D structures (membranes) and 3-D (rooms) cases are
usually treated separately.
As for excitations, these are usually not featured prominently, though
they are definitely there implicitly at least. I'd say in the papers that
I've read very often harmonic drivers (force-sustained) are assumed, but
not necessarily. It helps to bring the wave equation into reduced
Helmholtz form, which is convenient. It's a spatial problem only rather
than a temporal and spatial problem that way. In other formalisms, the
dynamic response in general usually with respect to the geometry of the
situation is considered in which case asymptotic arguments pop up (often
by lack of a better method). Asymptotic in this setting means that an
approximate form is assumed whose error shrinks with some parameter
becoming large, e.g. typically for high frequencies. Of course if the
situations could be treated directly, one would.
But despite all the simplications and reductions, the story isn't simple
(and not fully understood), which is I guess the point I wanted to make
with respect to the paragraph of the SciAm article.
- Georg
On Sat, 31 Jan 2004, beauchamp james w wrote:
> Date: Sat, 31 Jan 2004 09:38:22 -0600 (CST)
> From: beauchamp james w <jwbeauch@ux1.cso.uiuc.edu>
> To: gessl@CS.Princeton.EDU
> Cc: auditory@lists.mcgill.ca
> Subject: Re: reverse engineering of acoustic sources
>
> Dear Georg,
>
> Thank you for your wonderful response to my question.
>
> I wonder if any of the mathematical solutions to this problem take
> into account directivity and room responses and whether they work
> for forced sustained vibrations (e.g., clarinet) as opposed to
> free vibrations (e.g., a drum).
>
> Jim Beauchamp
>