On Fri, 16 Jun 2006, Cheng Cosine wrote:

Given a Helmholtz eqn with constant coefficients, assuming solution in
negative z-dir is neglegible, i.e., no reflecting wave in negative z-dir,
one can then use 2d FFT in xy-plane to solve the eqn. But when applying
the same argument to x- and y-dir, one can argue that wave still propogate
in positive and negative x- and y-dir since no simplifications are made in
those directions. That is, there are totally 5 propagation directions:
positive z-dir and neg/pos directions of x- and y-dirs. Then how can one
say that: it is a "plane wave in positive z-direction"?

Use separation of variables in Cartesian coordinates to find a general
solution to the Helmholtz equation. You end up with 2 arbitrary parameters
k_x and k_y which can vary from -infinity to +infinity for every
individual solution. Looking at the solutions, you can see that these
parameters are the x and y components of the wavevector.

Since you know the wavenumber, you can find k_z from

k^2 = k_x^2 + k_y^2 + k_z^2.

This gives you 2 distinct families of solutions: those where you take the
positive square root, and those where you take the negative square root.
These are plane waves propagating (at least partly) in the +z and -z
directions (except for k_z=0). Only when k_x=k_y=0 do the plane waves
propagate purely in the +z or -z directions.

The general solution is then an integral over some amplitude density
multiplied by these solutions, integrated over all k_x, k_y.

If for some given z=A, if all sources have zA, then you only have
non-zero contributions from modes with k_z0 for zA.

(That's assuming exp(ik.r-iwt) convention.)

Does this mean that one can neglect negative z-direction and then use FFT
to march along positive z-direction to solve ANY given wave sources without
errors?

No. A technical "no" because FFT isn't exact - you need the continuous
2D Fourier transformation in general.

A more general "no" because it isn't enough to know the sources - you need
to know the fields over an entire z=constant plane such that all of the
sources are on the -ve side of the plane. For the very restricted case of
a plane wave with wavevector in the +z direction incident on a screen with
some apertures, it's a very useful method. If you have a collection of
sources spread out through a 3D volume, you'll first need to find the
fields in a plane. In your post on sci.physics.electromag, you ask about
point sources. For point sources you can do this easily enough by using
the appropriate Green function. But then you don't need to bother with
Fourier transforms at all, since you can find the fields anywhere.

--
Timo Nieminen - Home page: http://www.physics.uq.edu.au/people/nieminen/
E-prints: http://eprint.uq.edu.au/view/person/...,_Timo_A..html
Shrine to Spirits: http://www.users.bigpond.com/timo_nieminen/spirits.html

"Timo Nieminen" wrote in message
news:Pine.LNX.4.50.0606161316310.9732-100000@localhost...
On Fri, 16 Jun 2006, Cheng Cosine wrote:

Given a Helmholtz eqn with constant coefficients, assuming solution in
negative z-dir is neglegible, i.e., no reflecting wave in negative z-dir,
one can then use 2d FFT in xy-plane to solve the eqn. But when applying
the same argument to x- and y-dir, one can argue that wave still
propogate
in positive and negative x- and y-dir since no simplifications are made
in
those directions. That is, there are totally 5 propagation directions:
positive z-dir and neg/pos directions of x- and y-dirs. Then how can one
say that: it is a "plane wave in positive z-direction"?

Use separation of variables in Cartesian coordinates to find a general
solution to the Helmholtz equation. You end up with 2 arbitrary parameters
k_x and k_y which can vary from -infinity to +infinity for every
individual solution. Looking at the solutions, you can see that these
parameters are the x and y components of the wavevector.

Since you know the wavenumber, you can find k_z from

k^2 = k_x^2 + k_y^2 + k_z^2.

This gives you 2 distinct families of solutions: those where you take the
positive square root, and those where you take the negative square root.
These are plane waves propagating (at least partly) in the +z and -z
directions (except for k_z=0). Only when k_x=k_y=0 do the plane waves
propagate purely in the +z or -z directions.

The general solution is then an integral over some amplitude density
multiplied by these solutions, integrated over all k_x, k_y.

If for some given z=A, if all sources have zA, then you only have
non-zero contributions from modes with k_z0 for zA.

(That's assuming exp(ik.r-iwt) convention.)

Does this mean that one can neglect negative z-direction and then use
FFT
to march along positive z-direction to solve ANY given wave sources
without
errors?

No. A technical "no" because FFT isn't exact - you need the continuous
2D Fourier transformation in general.

A more general "no" because it isn't enough to know the sources - you need
to know the fields over an entire z=constant plane such that all of the
sources are on the -ve side of the plane. For the very restricted case of
a plane wave with wavevector in the +z direction incident on a screen with
some apertures, it's a very useful method. If you have a collection of
sources spread out through a 3D volume, you'll first need to find the
fields in a plane. In your post on sci.physics.electromag, you ask about
point sources. For point sources you can do this easily enough by using
the appropriate Green function. But then you don't need to bother with
Fourier transforms at all, since you can find the fields anywhere.


Looks like the key point is to have the complex wave field in a z=const
plane, and then

one can march solution along a z-direction. For example, one has a complex
sound field

at z=const plane either from a single point source or from a set of point
sources distributing

in a 3D space, then one can use this approach to obtain solution at the
right side of z=const.

If one is interested in solution in negative z-direction, simply use the
same approach in the

other z-direction, right?

Now turn to Fourier transform and FFT. In many situations, one only has
measurement in

a finite plane, so only FFT can be performed. In this case, what are the
techniques to minimize

errors? Also, suppose the complex wave field at z=const plane is obtained
from a focused

source, say, waves emitted from a partial spherical surface, will this help
to reduce errors when

using FFT since most "features" or "information" are confined in a smaller
area?

Thanks,
by Cheng Cosine
Jun/16/2k6 NC

On Fri, 16 Jun 2006, Cheng Cosine wrote:

Looks like the key point is to have the complex wave field in a z=const
plane, and then

one can march solution along a z-direction.

Yes. As long as the sources are on the other side of the z=const plane.

For example, one has a complex
sound field

at z=const plane either from a single point source or from a set of point
sources distributing

in a 3D space, then one can use this approach to obtain solution at the
right side of z=const.

Yes, but if you just have point sources, you can directly find the
solution everywhere with much less effort. But as an illustration of the
method, sure.

If one is interested in solution in negative z-direction, simply use the
same approach in the

other z-direction, right?

On the other side of a different plane that's on the far side of the
sources. Two planes on which you know the fields, with the sources between
them, and you can use Fourier transform methods to find the solutions
anywhere in the region outside the space with the sources in it.

Now turn to Fourier transform and FFT. In many situations, one only has
measurement in

a finite plane, so only FFT can be performed.

2D FFT assumes that the fields are periodic. The result will be unreliable
after propagation over a sufficient distance for the wave in one finite
patch to diffract over the borders into neighboring patches.

In this case, what are the
techniques to minimize

errors? Also, suppose the complex wave field at z=const plane is obtained
from a focused

source, say, waves emitted from a partial spherical surface, will this help
to reduce errors when

using FFT since most "features" or "information" are confined in a smaller
area?

If you want to calculate propagation over a long distance, FFT is no good,
due to the above-mentioned diffractional spreading. It might be better to
assume that the field outside the patch is zero, and directly calculate
a discrete Fourier transform as an approximation of the continuous Fourier
transform. This will fail after a certain distance depending on the
spacing of discrete points (as opposed to FFT which will fail after a
distance depending on the size of the patch).

If the sources are contained in a spherical volume, so that you'll have
spherical waves, don't use plane wave solutions, use spherical wave
solutions. If you know the outgoing field over a spherical surface, then
you can find a spherical wave transform, and use that. Convergence
properties are supremely better than for plane waves - it will not fail at
any distance (because you get discrete modes, not a continuous set of
plane wave modes, and the convergence depends on the radius of a sphere
enclosing the sources).

I dealt with the reverse of this problem - spherical waves from outside
some volume converging to a focus. This was for electromagnetic waves, not
acoustic waves, but if you're interested, see:
T. A. Nieminen, H. Rubinsztein-Dunlop and N. R. Heckenberg
Multipole expansion of strongly focussed laser beams
J. Quant. Spect. Radiative Transfer 79-80, 1005-1017 (2003)
which is also on arxiv and available via eprints link below (when it
works, which is usually but not right now).

--
Timo Nieminen - Home page: http://www.physics.uq.edu.au/people/nieminen/
E-prints: http://eprint.uq.edu.au/view/person/...,_Timo_A..html
Shrine to Spirits: http://www.users.bigpond.com/timo_nieminen/spirits.html