wave function notation

Reposting in alt.sci.physics.acoustics and alt.sci.physics ...

"Andreas Håkansson" <[Only registered users see links. ].es> escribió en el mensaje
news:btj9a6$6kq$[Only registered users see links. ].upv.es...
included

Dear tadchem:

"tadchem" <[Only registered users see links. ]> wrote in message
news:[Only registered users see links. ]...
....

This could also be the cotangent of the argument, if any... not that this
observation would be any (big) help.

David A. Smith

"Andreas Håkansson" <[Only registered users see links. ].es> wrote in message
news:btk82p$pk2$[Only registered users see links. ].upv.es...
without

I recall once seeing something similar, where the writer's intent was to
represent that either function could be use to describe the observed
behavior, but I cannot recall exactly where or under what circumstances.

Practically, in acoustics, the vibration of a pure sound wave can be
represented equally well with either function with only a phase difference
[(2n+1)*pi/2] between the two.

Look to the _context_ to see what mathematical function works in the
particular equation. If you still have the article post the equation (if
you can) or a description of the context and we may be able to help you
further.


Tom Davidson
Richmond, VA

"N:dlzc D:aol T:com (dlzc)" <N: dlzc1 D:cox T:[Only registered users see links. ]> wrote in
message news:A8pLb.47894$gN.7810@fed1read05...

I had considered that possibility, but since the OP had mentioned that the
original article involved acoustic scattering, and the cotangent function
has regularly occurring singularities which would introduce difficulties in
acoustical descriptions, I discounted the possibility.


Tom Davidson
Richmond, VA

"tadchem" <[Only registered users see links. ]> escribió en el mensaje
news:[Only registered users see links. ]...

The article is the pioneer work of Waterman (J. of Acoustic Soc. of America
45, p1417) from 1968 on Scattering Matrix Method where he solves the
scattering porblem for a arbitrary scatterer by expressing the incident
wave as an expantion of wavefunctions expressed as the velocity potential
(i.e the solutions the scalar Helmholtz equattion):

Yn=const. * cos "over" sin (n*theta) * Hankel(n,k*r)
,where cos "over" sin is the mysterious part...

Further, what I'm really trying to do is implement the Multiple Scattering
Theory using the Scattering Matrix for non circular scatterers in 2D. So by
chance if there is an expert on the theme reading this please give me some
nice reference articles which I don't seem to be able to find by my self...

Andreas Håkansson
Valencia

>

I agree with Tom. It looks like both sin and cos are possible terms in this
solution to the wave equation in cylindrical coordinates. You could read
the equation above as two independent particular solutions. Morse and
Ingard use similar notation in for instance "Theoretical Acoustics" 1968 pg
215 when expressing the possible solutions of the oscillations of a plate.

Hal Zintel

I believe that you are referring to Eq. 33, [Waterman, "New
Formulation of Acoustic Scattering," J. Acoust. Soc. Amer. 45 no. 6,
pp. 1417 - 1429, 1969.

In this equation for the two-dimentional basis functions there are two
types of basis functions, even and odd that are denoted as "e" and
"o". In TeX notions, he writes
$$ \psi_{(e/o)} (\bf{r}) = (\eps_n)^{1 \over 2} { {\cos} \over {\sin}
} n\theta H_n (kr) $$
This single equation is really _two_ equations. One is
$$ \psi_e (\bf{r}) = ... \cos{ n\theta } ...$$
and the other is
$$ \psi_o (\bf{r}) = ... \sin{ \n\theta } ...$$
Using this, he then breaks up the Q matrix into "blocks" of even-even,
even-odd, odd-even and odd-odd submatrices. If the scatterer has
enough symmetry then the off-diagonal blocks will vanish. I hope that
this makes things clearer.

The Tmatrix formulation has been used extensively by Werby for
scattering from various ellipsoids. Werby has published extensively on
this in J. Acoust. Soc. Amer.

Trying to use this formulation with spherical basis functions may
bring convergence problems due to the fact that the expansion of the
surface field may not be sufficiently complete for some classes of
scattering shapes.


On Fri, 9 Jan 2004 10:54:37 +0100, "Andreas Håkansson"
<[Only registered users see links. ].es> wrote:

Thank you!
Though... I still have a question related to the notation in Watermans
article. I've been using Multiple Scattering based on the T-matrix method
for calculating the scattered field form a cluster of cylinders. The
incident and scattered base functions I've been using are the Bessel and
Hankel funktion just as in [Waterman]. Though instead of the term "cos over
sin n*theta" I have exp(i*n*theta) for both waves, incident and scattered.

Has this somthing to do with that I am using multiple scatterers instead of
just one?

And can I apply eq. 16* (in New Formulation of Acoustic Scattering) without
any modifications for these base functions?

..Andreas

"Charles F. Gaumond" <[Only registered users see links. ].mil> escribió en el mensaje
news:[Only registered users see links. ].navy.mil...
(if
America
Scattering
by
some
self...