On 2022-04-06, Fons Adriaensen wrote:
With near field sources, the outwards radiating field from a point source
is reactive at each point. Pressure and velocity are *not* in phase,
Even a pantophonic mic will pick that up that phase difference.
It will, but it will also miss out on energy radiating off the
pantophonic plane. Counterwise, a purely planar, pantophonic analysis in
reconstruction will also miss out on convergent energy from the rig.
This was in fact why Christoff Faller, when he lectured at Aalto
University in the day, thought "Ambisonics couldn't work". He took to
the fact that a pantophonic rig setup necessarily leads to a 1/r falloff
in amplitude as measured from the rim inwards. What he didn't understand
was that reconstructing that basic WXY-field utilizing a full periphonic
rig (or if you will an idealized cylindrical pantophonic setup,
reproducing not point but line sources) takes away the problem entirely.
Now, the counterpoint for a microphone is a bit different. A soundfield
mic as its central/coincident one doesn't have quite this problem, so
you can in theory do without Z, so as to have a coincident pantophonic
setup. At first order POA. That's because the acoustic field is
pointwise four-dimensional: pressure/W, and three velocity
components/XYZ, all independent. It's all orthogonal, so that you can
just drop Z with no harm done, energy or propagation direction wise.
Not so with anything above first order, because then you'll be dealing
with non-local features of the acoustic field. And once you do that,
parametrize it in direction, your Fourier series won't terminate. They
will attenuate by frequency, but they'll never truly terminate. Which is
why the first order Gerzonian soundfield is so easy to derive, while
even a second order mic actually necessarily is an engineering feat, and
nonperfect in its attenuation of higher order spherical harmonical
contributions. From second order up, you *cannot* just neglect the
vertical components, like you can with POA soundfields; it's the same
thing as it is with Faller's analysis of pantophonic playback rigs, and
their losing energy to the third dimension. As soon as you deal with a
field, non-locally, this sort of thing necessarily happens, and since
you can only deal with the acoustic field locally using four
coordinates, WXYZ, anything above that becomes more complicated.
This is even the reason the dominance/boost transformation of early yore
doesn't readily generalize. Because it's basically a local Lorenz boost,
akin to what is done in special relativity. It doesn't generalize to
non-local things, which is what anything beyond first order in
ambisonics necessarily is. It could generalize over if the acoustic
field pointwise had more degrees of freedom. Like, say, if it
*pointwise* had a second order spherical harmonical symmetry; if it was
a tensor field of that kind. But it isn't.
Also, below what we have in sound, it's also theoretically possible that
we couldn't have even this kind of extant symmetry. I mean, if acoustics
was described fully by a scalar W field, so that there was no velocity
part XYZ to its tensor, actually we couldn't have even a workable
soundfield mic. We'd have the same problems of spatial extension we have
with second order and further mics, starting with first order. Such it
is e.g. with the heat equation.
the vector describing energy transfer (in EM I think the Poynting vector)
is *not* in the plane, but outwards from the source, all round.
Which means that whatever you try to describe here is NOT a vector.
Of course it is. We might not yet agree where it points, but a vector it
is.
The velocity vector will be in the horizontal plane, and is
represented correctly by X,Y. It has no Z component.
This is not true. Suppose you have two point sources of sound, one metre
to your left and one to the right. They radiate 3D at an equal
amplitude, but also in opposite polarity.
When you vectorially sum their contributions at the origin, you'll
discover there is a vertical component. It's counter-intuitive to be
sure, but one way of seeing why it happens is that in such a nearfield,
the pressure field will be oscillating not just in the plane, but above
and below the center point. That then leads to an oscillating, vertical
pressure gradient, off plane, which forces transverse oscillation in
velocity as well.
This wouldn't happen if the sources were infinitely tall cylindrical
ones, since then there wouldn't be any pressure variation in the plane
of symmetry. Also, there wouldn't be any such phenomena if the equations
were 2D to begin with — though if they were, we also wouldn't have an
inverse *square* laws to deal with, but just a straight inverse.
But given even two monopole sources in 3D, in antiphony, at equal
distances, you indeed get lateral velocity/forces. Different kinds and
amounts too, depending on frequency and distance.
So what happens is that while the pressure field is fully symmetric
in the horizontal plane, there still has to be a Z component in order
to recreate the field to full first order.
Not for it to be correct for a listener in the same horizontal plane
as the speakers.
But yes. Faller's analysis was that you miss a 1/r component in
intensity by distance. The analysis I think holds as well. So pantophony
doesn't cut it. Even if you want to recreate a well-recorded pantophonic
field, you actually need to recreate it using periphony, in order to
avoid that 1/r fading in amplitude.
Of course if the listener moves up or down the sound field he/she
senses will be incorrect. What do you expect?
Well that'd just be stupid. What do you expect.
No, no, I'm talking about a fully distinct phenomenon. About something
far more interesting and intricate.
Or if I'm perchance poking holes in my head, at least I'm doing so in
good company. Analysing stuff rationally, instead of just shouting
into the wind. So do poke me. Let's see where this gets us. :)
...and that's precisely why pantophony is an idea born dead. We don't
have infinite vertical line sources, nor microphone arrays which
mimic their directional patterns. The only thing we really have is 3D
mic arrays and 3D rigs.
Indeed. But we also have situations in which most sources are in the
horizontal plane or close to it, and as listeners we tend to stay on
the ground and not fly around.
Then I think the most interesting thing is to adapt our mic arrays *to*
this situation. While we also adapt our mathematical machinery to it as
well. Our reconstruction machinery.
The HOA machinery, WFS too, is pretty good at analysing what then
happens. Only, they haven't much been used to deal with such uneven,
anisotropic kinds of problems.
Maybe we/someone ought to take the theory towards those kinds of
problems as well?
--
Sampo Syreeni, aka decoy - de...@iki.fi, http://decoy.iki.fi/front
+358-40-3751464, 025E D175 ABE5 027C 9494 EEB0 E090 8BA9 0509 85C2
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