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

[Moses Fayngold <moshfarlan@yahoo.com>]

- On Sun, 2/28/10, John Denker <jsd@av8n.com> wrote:

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.
Flat denial is not an argument. My original questions have not been answered. 

> > 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.

  By (temporal!) frequency omega I meant number of cycles per second (times 2*pi). A plane wave has BOTH - a frequency and propagation vector k. If you call its MAGNITUDE |k| the spatial frequency, it's fine. A spherical wave with frequency omega is equivalent to a continuous set of plane waves with the same frequency omega but different propagation directions, according to dispersion equation (kx)^2 + (ky)^2 +(kz)^2 = (|k|/u)^2, where u is the phase velocity.   Once omega is fixed, we have the stationary state. In case of wave motion it is described by the Helmholtz equation, with the temporal dependence eliminated. In case of spherical symmetry, you can get the solution by solving the equation in spherical coordinates to get const* sin(|k|*r)/r; alternatively, you can write the int{C(k)*exp(ik.r)dk}. For |k| fixed it reduces to the integral over the surface of radius |k| in the 3-vector space, and for spherical symmetry we have that
 Fourier-amplitudes C(k) = const for all directions of k. In this case the integral can be easily taken and you will get the same expression const*sin(|k|*r)/r. This is a standing spherical wave (the simplest example of Bessel's spherical function). It is standing because in the original integral to each plane wave propagating in some direction there is its counterpart propagating in the opposite direction. You got the radial dependence of the amplitude which reflects, as you said correctly in a previous message, conservation of energy. But it has nothing to do with dispersion. Even in a dispersive medium there cannot be any dispersion when all your waves have the same frequency and wavelength. And the spherical wave in my example is represented through the Fourier transform of the plane waves - all with the same frequency, be it temporal or spatial frequency.   Note that the solution we get from the Helmholtz equation is a superposition of diverging
 and converging MONOCHROMATIC spherical waves - it is necessary for a stationary state.   I cannot do the same trick with diverging wave alone (restricting to fixed |k|) because such state would not be stationary; here we have a beautiful analogy with the theory of radioactive decay. But I can do this trick if I introduce the source - the equation will become inhomogeneous. In a way, the converging wave plays the role of the source - it maintains the diverging one. The same can be said if we replace converging spherical wave by additional plane wave - then we will have the scattering problem (in a stationary state formulation).   Now we can do the same procedure with another frequency etc. and add up the results.Suppose that each constituting plane wave is converted by this procedure into a sharp spike propagating perpendicular to its plane. Now we have a superposition (truly 3-D Fourier integral) of plane spikes (with identical Fourier
 transforms!) propagating in all directions; and the outcome would be... a spherical shell with the thickness of the spike. If each spike spreads (there is dispersion) so will the shell (its thickness will increase). But if there is no dispersion and each plane spike propagates without changing its shape, their sum - the spherical shell - will expand with the speed u = const without changing its thickness. If you have two plane spikes of the same spectral composition moving in different directions in a LINEAR medium, they will not start spreading only because they intersect with one another (linear superposition cannot change the shape of the intersection line). If I am wrong, and pure change of shape of the wave front can produce dispersive spreading, this must be true for the EM waves in vacuum as well. Each monochromatic component of such wave is described by the same Helmholtz equation with u = c (polarization neglected). Identical equations have
 identical solutions. If it is true that mere converting of a plane wave into spherical is sufficient to produce dispersive spreading, then scattering of a MONOCHROMATIC plane wave on a spherical object (or just passing it through a small hole in an opaque screen) would produce a SPECTRUM of diverging spherical weaves. And vice versa, scattering of a plane sharp laser pulse (originally moving, as any decent EM perturbation in vacuum does, without spreading!) on a small obstacle would produce a luminous shell which is not just expanding, but also SPREADING in its width, which means that its monochromatic components all of a sudden decided to move each with its individual speed. My initial statement (a mere change of wave-fronts' shape in the same space cannot by itself affect the dispersion) is equivalent to the the ban on this nightmare. If my statement was wrong, as JD insists, this nightmare would be the reality.   I have nothing more to add to this.
            
> 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 does not add anything to this discussion, since it is self-evident that periodic source (in the time domain) produces periodic response. This was known long before Floquet. This theorem and the corresponding solutions (the Bloch functions) are important in spatial domain of space-time when there is a spatially-periodic potential, but this is totally irrelevant to our topic since we consider waves in a homogeneous medium.  

> 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. 
  True, but Irrelevant to our topic
> The system has a symmetry, but it>is not a translational symmetry.  
Actually, it depends on the way we do spatial translation. If we displace everything - the source and the origin of coordinate system - by the same amount in the same direction, the physics will not change as long as the space itself is homogeneous, and this is all that matters in our discussion. 
> It is, well, a radial symmetry.
  Sorry, I am not familiar with this term. I would rather say, a rotational symmetry about any axis passing through the origin. And we use this symmetry when composing the symmetrical Fourier transform of the type described above. 

> Also keep in mind that it is bad luck to "prove" things that
> aren't true. 
  100% 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

I did not find anything in this link except for description of the tank. 
> 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

 I have already commented on this before. In 1-D, the dispersion equation reduces to k = omega/u, so k*x - omega*t becomes 
                      k*(x - u(k)*t) = omega*((x/u(omega)) - t),                         (1)
and integration over k is automatically equivalent to integration over frequency. The system in the last two links is innately non-monochromatic in the sense of Eq. (1). No surprise it is spread in the frequency domain. It does not add anything to our discussion.
Moses Fayngold,NJIT  


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