Eckard,
You have condensed quite a lot, into two brief paragraphs! Where do I begin!?
The point you make is that: realistically it should be obvious that the vertical bar of the N is the origin in the time parameter of the physical wave which we measure. I agree absolutely. It is in how we can measure anything experimentally that makes it all fuzzy, because what we get as the result is physically the response in the field of a detection device, and can only surmise from what we do know without presuming on what we do not know, what constitutes the real physicality of the wave itself. In fluid hydrodynamics, such as acoustics, we can at least observe something of a medium through which the wave propagates. BUT, we can only use that as an approximation of the physical dynamics of an electromagnetic wave which propagates on its own across free space devoid of any medium. And which when encountering any sort of medium, interacts in several ways to either be absorbed, refracted or reflected. Allow me to start by distinguishing that we can only allow that a planar traverse wave is applicable to what would correlate to the surface response of fluid such as water.
If we are observing the translation of a wave from an aspect perpendicular to the longitudinal direction of propagation, the sinusoidal shape of the wave is evident as the signature of a rise and fall of effective response in the detector. If we choose the wave to be a planar transverse wave, that sinusoid is in reality a cross-section representation. But, we can rotate our position around that longitudinal direction while remaining perpendicular to it as if we were (as observer) orbiting in a spy capsule. And the vertical bar of origin would appear unaltered. The time parameter is real, but our choice of measure origin is still observer dependent, unless and only if the wave is a true hydrodynamic surface planar traverse wave. A three dimensional dynamic wave event, whether an air blast or electromagnetic emission would still register a signature as a sinusoid regardless of our position in orbit. Physically the traverse nature of the wave signature is observer dependent and not the total reality.
As to "focusing on the astonishing relations between sinusoidal and exponential functions." that relationship has the form of a hyperbolic function independent of variable parameters. A hyperbola is distinguished as a conical section where the plane is NOT parallel to the slope of the side of the cone, and so the 'legs' of the hyperbola diverge, geometrically. A natural exponential function from which we obtain the base of natural logarithms, can relate to cosine transformations in the rate of change of slope on a sinusoidal curve.
As the observed *N* shape morphs across the time parameter into a bell shape, and a continuous repetition of that bell curve propagates, we observe the sinusoidal wave 'train' of response in our detector which operates as a hyperbolic function. UNLESS!....
...what our detector responds to results in an output of a photoelectric effect! A kinetic, not electromagnetic response. Just as if I were to stroke a pool cue ball to collide with an object ball and the rate of transfer of momentum to the object ball would plot as a ballistic curve, just like the trajectory of a cannonball falling in a gravitational field. And a ballistic curve is a parabolic function, not hyperbolic. A parabola is a conical section where the plane IS parallel to the slope of the side of the cone, and so the 'legs' of the parabola converge towards being parallel with each other. A bell shaped curve cannot evolve to be repeated! AND, while both hyperbolic and parabolic functions have a mathematical identity as *exponential functions*, axiomatically they are not of the same kind. A parabolic function extrapolates from the harmonic series of 2n, and the acceleration of gravity and the half-life of radiological decay are found on the parabolic plane. A hyperbolic function extrapolates from the exponential series of 1+1/n and while plotting close to the harmonic series, has a slightly different rate of change of slope.
So while Fourier Analysis is highly effective in filtering, not 'out' but 'down', to a recognizable wave signal, we are still far from being able to finitely measure a single quanta in an observed wave event. In fact, we really can't say definitely that an electromagnetic emission is a physical wave itself rather than it being a linear projection of energy which is reacted to by detection devices whose inherent energy transfer results in a time dependent rise and fall of induced reactance.
But those are our clues. How is it that an observed kinetic impact (parabolic) can initiate an electromagnetic emission (hyperbolic) such as a piezo-electric oscillator, and an electromagnetic absorption produce a kinetic response. And what determines whether a hyperbolic event, or parabolic event, propagates into yet another of its axiomatic kind, or rotates from a hyperbolic plane onto a parabolic plane, or vise versa.
We are still stuck on Wave-Particle Duality. We must be ignoring something obvious. :-) jrc