A basic but interesting fact:
At the
point on a dispersion diagram, where
and
, the usual phase velocity is undefined; that is,
.
So, what do we do when this occurs? We use L’Hospital’s rule. This yields 
When 
I’ll go ahead and sling this out… a realization is that the group velocity, (effective)mass, etc. lose significance when spatiotemporal “texture” disappears.
I often think about how our world reflects us, and vice versa. For instance, consider the reciprocity inherent in mass media, and how we might effectively look through the TV screen and at a blurry image of ourselves: society at-large. It’s “media as our looking glass.”
Advertisements during Oprah or The View are very different from those during football, Glee, Wipeout … prime time. Marketing companies have intently studied the viewership, and so the ads are targeted to have greatest return. I can reasonably suspect that an above average number of incontinent adults, or adult diaper-wearers (hmm, or maybe adult diaper purchasers), etc. are sitting in front of TVs watching “The View” at that moment when the commercial airs.
Today I typed “how do” into Google, and its relatively new “instant” feature displays the top few search queries, I suppose the few that are most prevalent (maybe even my suspected locale, “Peachtree City” on the left sidebar). It appears that Googlers most commonly want to know two things: how facebook works, and how to get pregnant. I guess you still can’t actually get pregnant by facebook, but maybe that’s a market-ready next step.
Google searches beginning with "how do"
Google searches beginning with "why do b"
I’m not sure what else Google is going to tell me, and I’m not really sure exactly what it means, but I’m really curious about people self-educating through the web.
In a group presentation I gave this week, I showed the math (now improved with a few corrections) for characteristics of a wave equation in general media (i.e. over a periodic basis). It is well known that Fourier techniques can be applied in partial differential equations to represent distributions of concentration, heat, electric charge, etc. This is especially appropriate for crystalline materials with well-defined periodic structure, but it is extensible to general media.
The first step was to reduce Newton’s 2nd Law from tensor form to a 1-D form, i.e. the propagation of longitudinal waves along x. (A nice feature of acoustical treatments in general, is that its formulation permits waves of general polarization, including longitudinal. Electromagnetic waves do not, except in the presence of media which can impart phase front tilt.)
I’ll eventually give the math here (and hopefully it will pass muster in my thesis), but it turns out that spatial periodicity of the material stiffness gives rise to a complex (mathematical sense) wave equation having orthogonal components. It represents a coupled heat equation and wave equation, where the dissipation term is converse to our usual expectations; it is (without coefficients):
Now there are several details and other terms which appear, but this snippet captures a form which many might find surprising. After visualizing the spatially periodic characteristics of the general PDE, I believe this has improved my understanding of dispersion diagrams: why they are generally complex (again, mathematical sense), and how we design devices using “meta” materials, and PxCs.
This investigation of periodicity-induced modal diffusion may only be novel to me, but I just haven’t seen this phenomenon described in simple terms before. See the references for some particularly impenetrable notation describing dispersion and mode coupling … but no explicit mention of inherent diffusion equation(s).
Last year, NPR’s Science Friday marked SETI’s 25th birthday. This sparked a few thoughts…
What if we’ve already got evidence of “extraterrestrial intelligence” in every signal we observe?
How would contemporary science explain that? This idea stemmed from consideration of the signal characteristics I would expect to receive from a super-intelligent being or civilization:
Are we all extraterrestrials?
OK, that’s a weird statement; but surely I am not here on this planet of my own plan & design. Explaining, would some scientists agree that are we all created by an unknowable universal energy (information) source? The biomolecular chemistry which enables terrestrial life at the cellular level is powered by the most natural and universal behaviors of stochastic motion, diffusion, and equilibrium. We are fundamentally driven by signals of indeterminate origin and assumed to be “noise.” From such stochastic processes we can obtain “order” in the limit of distributions and averages.
Contemporary scientific theories suggest that all the structure we perceive originated in a singular event of pure (scalar) power comprising everything: time, space, etc. This creation singularity, maybe mapped through the 
One might ask why we don’t see evidence of life elsewhere… perhaps that is because life is fractal, and our very measurements have their bases within that fractal, making observable things in some sense self-similar. We know measurements to be inner products, and thus, projections of the world onto the world around us. This also relates to mutual information in communications theory: a source and receiver should speak the same language, having that information in common. But, I suspect, thanks to an interplay of heat and waves, that the fractal has a “forward” direction in which we find new variations on old observations.
And I can’t help but note: this whole “extraterrestrial” creator thing seems like a secular perspective of God – that which cannot be seen but is omnipresent and ultimately responsible for all phenomena. We as humans perceive only things our amazing but limited senses permit. Yet we do find new perspectives – rearrangements of state – and this might be likened to looking through a kaleidoscope.
And of course I have comments on the expert commentary during the Science Friday broadcast:
Group theoretic expressions of optical singularities in photonic crystals
by Wheeldon, Jeffrey F., Ph.D., University of Ottawa (Canada), 2009, 268 pages; AAT NR51824
(PDF costs 37.00 USD for members of academic institutions.)
I contacted Jeffrey Wheeldon, and he seemed interested that his work could be relevant in that way. However, I’m not sure anything will come of the connection. Regardless, it so happens that Maxwell actually published a “molecular vortex model” which was later refined in his “On the Physical Lines of Force,” so the father of E/M has already opened the door.
This resonates with several viewpoints I sometimes think about: the wave structure of matter (atoms from waves), the conveyance of angular momentum (photons) among singularities (hearts of atoms), and others.
So I pose a question (thesis topic anyone?): “Would a 3-D extension of Wheeldon’s 2-D treatment of singularities in linear photonic crystals correspond to a structure of matter?” The scope would require much refinement.
While the claim may be a stretch for anyone, perhaps even Dr. Wheeldon, the applicability of group theory to the conditions on optical vortices in in photonic crystals just screams “standard model” to me. We shall see…
