Rotational waves

 

Lord Rayleigh ( John Strutt) made some influential notes about wave motion throughout his life. Bearing in mind he was born just before Quaternions were announced and Grassmann published his Ausdehnungslehre to a dismal response, and was in university at Cambridge about the time Maxwell published on Electromagnetism, using Quaternions and MacCullaghs curl potential, we can see he was right in the thick of the rests early attempts to model 3 d rotation mathematically.

 

It was really down to a few doughty souls to progress physics of the wave to its prominent position vis a vis the corpuscular dynamics of chemistry, which was making noteable headway in the industrial setting.

 

We have seen how Arago and Fresnel created a huge rift, with young , in the philosophical explanation of matter in the aether or plenum. While Newton provided a consisten theoretical model based on corpuscles , it was evident that it was not physical or empirical. At the same time the Wave theory was not physical with regard to light. Youngs experimental double slit interference patterns were not convincing enough , and it was the influence of Fresnel and Arago that enabled the results to make headway in the broader scientific, non chemistry based community. These tended to be more mathematically minded scientists who could understand the sine  graph, intruded by Euler as a model of a wave.

 

The notion of a wave is very rarely examined. One is usually immediately programmed to consider the circular functions of Euler as a wave. Thus a disconnect with physicality is immediately taught. Scientists no longer see any real wave, but rather approximations to the ideal sine graph! However in this process the ideal sine graph is misconstrued as a wave and so it’s true meaning is lost even as it is plainly laid out before the students eyes.

 

Firstly let us remove the blinkers. 

 

Euler took a circle of unit radius, that is its radius was defined as 1. Then he defined it’s semi circle or hemi arc as [tex]\pi[/tex] to about 30 decimal places. Thus he was able to draw an axis marked off in units of [tex]pi[/tex]. Thus this axis represented the rotation of a point around the circle or the motion of the centre as the circle rolled in that axial direction . In each case the circle was in dynamic motion called rotation.

 

Thus the sine graph represents not a wave motion , whatever that may be , but a rotation motion.

 

Now let us turn to wave motion. It must be observed that wave motion, vibration and periodicity are tautologically the same perceived behaviours. Any difference lies in the observers intention or purposes. Thus in the context of a sea wave the perception of a rolling body of water traversing the surface of the sea and rolling out onto the beach gives way to the undulatory motion of such waves on the personal stability of the observer. Indeed the bobbing motion of floating objects predominates over the passage of a rolling wad of water beneath ! 

 

Waves are observable on the surface of flats flowing rivers, but there the current predominates the observers senses and little mention is made of them. So what are the causes of these mounds of water in the surface of a dynamic fluid? It turned out not to be bobbing at all , but complex vortex behaviour. Both Lord Kelvin and Helmholtz regarded this as a groundbreaking phenomenon and they set out to describe a kinematics of vorticity. A first attempt.

 

This was a major influence on Stokes, Navier and Rayleigh, but Maxwell was conceptually in advance of these 2 great mathematical physicists. He wanted the vortices to act like gears nd springs and transmit strain. He opted to use Hamiltons Quaternions to express his ideas. Lord Kelvin was not amused. He like many scientists in his time felt this use of the imaginaries was Jabberwokky. A term coined by Lewis Carol, a prominent traditional Mathematicin, who derided this kind of Alice in wonderland mathematics in his book of the same title.

 

Consequently Maxwell was forced to recent, and in a remarkable turn around went from prise of Quaternions to a dire denouncing of them! This was at the behest of Lord Kelvin who was developing the ideas of vectors set out by a young American student of thermodynamics called Gibbs. It is a dark but not unfamiliar tale of underhand tactics. As a result, overnight research into Quaternions was shelved in America after a fateful conference on the issue of how physics should be taught. 

 

Maxwells statistical approach to gases suited Lord Kelvins own Kinetic theory and so statistical Mrchanics was developed by Gibbs to great effect, but the mathematics of fluid mechanics and ths Elrctromagnetism based on that floundered. This was because Maxwell expressed all the main concepts in terms of Quaternions. The fledgling vector algebras were not sufficiently graped to be able to compete with this elegant description. In addition, the Curl of a vector field was developed by McCullagh a mathematician in the same tradition as Hamilton, who used Quaternions to formulate his ideas, and the relationship with Knots and the properties of vortices in space.

 

The second tautological concept of a wave is periodicity. Thus when we experience the unwise everyday we apprehend periodicity, but hardly intend to call it a wave! It is clearly a rotation which involves very large scales of distance and time. Nevertheless we have to cknoledge that repeated variation which immediately makes it sn logos to regular bobbing up and down as in wave motion.

 

Periodicity reveals to me the essential rotation that is evident in a sea wave is lo evident at a much larger scale in astronomical terms. Astronomers since Eudoxus have modelled these circular motions to give. Apparent relative motions of planets. These motions were very wavelike and hence planets were called wanderers!

 

We now know that our solar  system wanders in the milky way galaxy on some spiralling rotating arm of the galactic structure. This wavelike motion is on a time scale of tens of thousands of years and on a displacement on sn astronomical scale .

 

My third example of the notion of wave motion is vibration. Typically we think of a piano string or a washing machine . We are told to think a piano string vibrates up and down. In fact it vibrates round and round! Despite precise plucking or striking the mechanical behaviour of taught wires in vibration is rotational. These rotations may be elliptical rather than circular but they are not up and down like a slow moving tension curl in a skipping rope.

 

While it is always possible to dampen the elliptical motion ofa vibrating string by placing constraints, this only emphasises the point. Vibrations are helical waves travelling bidirectionally in a tensile medium.

 

It really does not matter what scale you go to vibration or wave motion is due to rotational motion .

 

It is clear that rotation at any scale is almost similar. Thus we can expect the same mathematical formulae for wave motion to apply at ll scales.

 

Schroedinger’s wave equation is simply derived for rotating systems at ll scales. The idea that an atom is a planetary system look alike makedps this expectation almost inevitable. However we must not confuse rotation with planetary systems. A much more general graph of a rolling circle is called a trochoid.mit is complexes of these that better describe arbitrary rotation in space. We shall see that means regionality is inherent in rotational motion, as is integer relationships between regional complexes.

 

These regional complexes define a fractal Geometry and a fractal distribution

 

<From Fractalforums.com topic Light page 11>

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Why Rayleigh waves are the general wave wave mechanics

https://youtu.be/LCICoR6czcc

This is a 2d Rayleigh wave. We will see how a Rayleigh wave is the general wave notion we should use in physics and how it avoids the ultraviolet catastrophe.

https://youtu.be/ny2cKPGQf6Y

https://youtu.be/kb0lvi4L9No

This is the math that Rayleigh used with a colleague to derive the ultraviolet catastrophic equation. Notice it does not use Rayleigh waves.

The Planck derivation is in the first video

https://youtu.be/7yPTa8qi5X8

A clearer depiction of a Rayleigh wave , evident at a surface boundary where freedom of movement I greater.

https://youtu.be/wiSaNwk6BR4

So we see the the many modes of wave propagation are generalised in the spherical wave with an exponential description. At any surface boundary love and Rayleigh waves are generated.

Assuming a sine wave node at a black body boundary is thus a mistake, Planck by assuming a spring behaviour was closer to boundary reality conditions.

The propagation beyond the boundary , the black body glow will therefore be a spherical wave propagation. Such a propagation necessarily occurs in discrete frequency modes which Planck called quanta without understanding why his spring model was better than Rayleigh node model.

https://youtu.be/13hEfEa610A

https://youtu.be/Zbc8ZOiWBPs

How do we go from springs to probability?

The answer is interesting and illuminating.

De Moivre originated probability theory. De Moivre, sir Roger Coates and Newton were a powerful research team exploring Newton’s ideas and concepts. Newton was a master of infinite series and of the math of the geometry of the unit circle. Consequently he was able to accept and use the negative quantities of Brahmagupta and the other Indian mathematicians along with the Algebraic concepts of Bombelli regarding the square root of -1. .he was a student of Isaac Barrow and learnt Pythagorean geometry through his influence. He consequently formed and solved many multinomial equations and set out a basis for polynomial theory. His understanding of difference equations and expressions in calculating the trigonometric and natural and Briggs Logarithmic tables was unexcelled by his peers, . He took on De Moivre as a personal disciple and later sir Roger Coates. Both Coates and De Moivre collaborated on the theory of roots of unity in other words the discretisation of the unit circle. . Coates went on to propose the Coates Euler theorem in its logarithmic form decades before Euler proposed it in his exponential form.. of course he died before he could explain it to Newton as a generalisation of Newton’s force laws . Later Boscovich completed this line of research , really establishing the force relations as Fourier type expressions’ which are based on rotational dynamics.

However DeMoivre took the unit circle ideas in the direction of establishing probability theory which he.did , barely establishing preeminence over another developer.

◦ These ideas of probability or expected outcomes were applied by Boltzmann to physical phenomena where populations of agents could reasonably be expected. Needless to say it was not popular with his peers who wanted exact or precise solutions. However Gauss showed how by using these ideas he could precisely predict the orbit of a comet which was only sporadically glimpsed and often immaculately measured. . By applying Boltzmann normal distribution curve to these varying data he was able to determine a bell shaped distribution which gave the probability of a range of measurements. , meanwhile Fourier was demonstrating how trigonometric functions could interpolate any polynomial and indeed any curve shape. Thus a population of sine or cosi e functions could describe a set of experimental data . Lord Kelvin promoted this view as the way forward in physics especially for the growing molecular description of material behaviour, At the same time Maxwell was using the Gauss Boltzmann normal distribution probabilities to characterise gases and the velocity of their Dalton molecular structure. All at that time accepted the aether continuum and so vortices in a continuum as proved by Helmholtz became the favoured corpuscular model for the atom. . There were no electrons until JJ Thompson demonstrated a ratio Metternich mass and so called EMF which could just as well be explained as magnetic induction force given a dynamic magnetic vortex.

So Rayleigh in relating radiation energy to frequency assumed a purely sinusoids wave. . Such a wave as a standing wave can have any frequency that fits the cavity. . In his reasoning the frequency obscured the wavelength. We can see that only half wavelengths can be counted, but not all frequencies will have half wave lengths that will fit a cavity. These were assumed to be destructively cancelled. . So right there we have discretised frequencies. However by averaging they obscured this discrete condition.

Planck on the other hand was considering populations of springs . He therefore had little choice than to start with a generalised Fourier description with its discrete frequencies. . Using the Maxwell Gauss Boltzmann De Moivre ideas an accepted energy probability curve was found, . He could not average away the discrete frequencirs. Instead he had to use the standard series sums to simplify and this gave a different form to the description. It also tied in his results with the different series used to depict the wavelength lines in rhe spectral analysis of emission spectra.

We see that classical exact formulations were not up to the job, but classical probability theory and Fourier analysis were.

Ironically Rayleigh himself pointed to flaws in the wave mechanics of his day relying too heavily on the simple sine wave. He expressed his wave mechanics in the complex Fourier form, and so predicted Rayleigh waves.Love then predicted love waves. But Planck by focusing on the modes of spring oscillations unwittingly uncovere the physical explanation of these modes of oscillation and the quanta required to isolate them.

Quanta are interesting. Because we do not understand the arithmoi we fail to grasp that quanta or units have to be carefully distinguished. So quanta in relativity are units of space times time. In rotational dynamics they are u its of h times frequency. In Newton’s principles quanta are units of density times volume and J J Thompson discovered a quantum that is units of mass times deposition time, called the EMF. Both electric and magnetic induction are used to establish this quantum which is why we call such waves electromagnetic.

https://youtu.be/ErRhupNgFS8