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[Phys-l] Floquet's Theorem ... or not



On 02/24/2010 09:31 AM, Moses Fayngold wrote:

... Whether a wave (more accvurately, its
propagation) is dispersive or not is determined by the properties of
the medium and, if we also take into account the non-linear terms, by
the wave amplitude, but not by geometry.

That is untrue. It was untrue last Wednesday, and it
will be untrue next week and forever after.

For each frequency, a spherical wave can be represented as a linear
superposition (3-D Fourier transform) of plane waves with the same
frequency.

What's true in the time direction is not true in the
spatial direction(s). And since this refers to a 3D Fourier
transform, I must assume that the "frequency" here is a
spatial frequency.

Floquet's theorem guarantees that if we drive this system with
a force that is periodic _in t_ then the response will be periodic
_in t_ (with the same period).

Floquet's theorem didn't just pop out of nowhere. There are
significant provisos attached to it. In particular, we must
insist that the system is linear and that its response (in
particular, its Green function) is invariant with respect to
shifts.

This is true for our system provided we consider only shifts
_in the t direction_.

Alas, a radially-symmetric system, which is the type of geometry
we are considering, is not invariant with respect to shifts in
the x, y, z, or r directions. The system has a symmetry, but it
is not a translational symmetry. It is, well, a radial symmetry.

You can take the spatial Fourier transform as suggested above,
but the result is a mess. The result has energy at infinitely
many spatial frequencies. The system is still periodic _in t_.
And if we consider only the far field, it can be locally and
approximately treated as a superposition of plane waves, but
this is nowhere near being a global solution ... and it is nowhere
near being a proof (or even evidence) that the geometry doesn't
matter. If you have a method of solution that works only for the
temporal part of the wavefunction, but not for the spatial part,
then you don't have a solution.

Also keep in mind that it is bad luck to "prove" things that
aren't true. It is well known that propagation is highly
dispersive in D=2 (cylindrical) geometry, even in a medium
where plane waves would be non-dispersive. You can do this
experiment in a PSSC-style ripple tank.
http://americanhistory.si.edu/collections/object.cfm?key=35&objkey=5894

It's also really easy to do the calculation. Here's a snapshot
of the spatial wave function with bilateral symmetry.
http://www.av8n.com/physics/img48/radial_symmetry-xy.png
And here is the Fourier transform thereof:
http://www.av8n.com/physics/img48/radial_symmetry-hires.png

As expected, it's a mess. There is energy in lots of places
other than the nominal (spatial) frequency.