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These concepts of Rayleigh Love Transverse and longitudinal waves explain where we go wrong. Their is no pure transverse wave dynamic! The viscosity and relaxation characteristics of materiality are a combining term which makes all so called wave propagation a Rayleigh or love wave, even in so called gaseous fluids. The ratio of transverse to longitudinal varies and is a characteristic of the medium as well as its phase and phase structure. Thus a plasma is typically conceived as a multiple phase medium consisting of “particles” of differing volume and magnetic characteristics. Thegasous phase is mixed with the fluid and solid phases and given a purported “ electric” charge structural dynamic . We could reconsider the whole plasma to be a fractally distributed ensemble of magnetic regions of varying densities/intensities through which Rayleigh and love waves pass as currents .
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.
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
A clearer depiction of a Rayleigh wave , evident at a surface boundary where freedom of movement I greater.
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.
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.
It was really down to a few doughty souls to progress physics of the wave to its prominent position vis a vis yhe 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 Newyon 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 Gometry and a fractal distribution
Why miss out the magnetic current JJ Thompson used to balance th system?
Why was the space in the plum pudding empty? Changing it to a solar systmmodel imports the fallacy of empty spac!w know space is full of magnetic current and plasma .. is the particle even asolid or a density as Newton posited?
Why miss out Planck and his springs? Why ignore Rayleigh notes on wave mechanics ? Why ignore that the schroedinger wave equation applies equally to planetary orbits and moon orbits around planets? Why miss out probability and statistical interpretation of data?
Starting with magnetic current as a basis gives a magneto dynamic explanation for the elemental structures and fractally distributed regions of density plus absorption and re transmission of magnetic patterns in a trochoidal dynamic. . Because probability is derived from rotation in the unit circle it can describe pattern outcomes for experiments in the magnetic aether. .
The so called complex quantities are simply rotational or curvilinear arc magnitudes for general spirals.
Does the electron spiral into the centre? Yes if we allow it or a reaction like it to spiral out again!
We can consider radiation as equivalent to mass ejection of the density core of a fractal region when the spiral equilibrium is disturbed. The plum pudding is not empty , it is full of magnetic current behaviours in my opinion and in the model I promote.
Even on a planetary scale we have evidence of magnetic current effects on surface Rayleigh and love wave phenomena .
Why miss out the magnetic current JJ Thompson used to balance th system?
Why was the space in the plum pudding empty? Changing it to a solar systmmodel imports the fallacy of empty spac!w know space is full of magnetic current and plasma .. is the particle even asolid or a density as Newton posited?
Why miss out Planck and his springs? Why ignore Rayleigh notes on wave mechanics ? Why ignore that the schroedinger wave equation applies equally to planetary orbits and moon orbits around planets? Why miss out probability and statistical interpretation of data?
Starting with magnetic current as a basis gives a magneto dynamic explanation for the elemental structures and fractally distributed regions of density plus absorption and re transmission of magnetic patterns in a trochoidal dynamic. . Because probability is derived from rotation in the unit circle it can describe pattern outcomes for experiments in the magnetic aether. .
The so called complex quantities are simply rotational or curvilinear arc magnitudes for general spirals.
Does the electron spiral into the centre? Yes if we allow it or a reaction like it to spiral out again!
We can consider radiation as equivalent to mass ejection of the density core of a fractal region when the spiral equilibrium is disturbed. The plum pudding is not empty , it is full of magnetic current behaviours in my opinion and in the model I promote.
Even on a planetary scale we have evidence of magnetic current effects on surface Rayleigh and love wave phenomena .
Ok so start with this explanation, realise that the sine wave is misleading because it only contains transverse information and it ignores trochoidal dynamics, so it needs to be put in the exponential form, and that involved using an imaginary quantity “i ” and so Dirac obscured that with his BraKet notation .i is a quarter turn magnitude and the algebraic process constructed for it by mathmagicians involves rotating a vector of unit length around an unspecified centre. . 2i refers to doubling the length of the vector after rotating , i + i means the same thing while i ^2 means a quarter turn followed by another quarter turn. The numeral in front of i is the scalar applied after the quarter turn. .
The number line is often added to these magnitudes. In that sense the number line only gives a numerical adjunct, however if we regard the number line as a fixed vector ,one , then we can represent rotations around fixed positions in space , and subsequently around dynamic positions in space. . We have to accept the planar nature of these rotations, but in combination the represent spheroid always, but more generally spiral or vortex ,trochoidal rotations.
Missing from the description of light is the magnetic fundamental behaviour: the trochoidal rotation. So to assume electricity generates magnetism by a current is in my opinion”arse over tit” the magnetic dynamic is a magnetic current that is fractally trochoidal. This generates all so called electric behaviour in the electric phase and frequency range and the further ranges of both infra and ultra descriptors and beyond..
We are helpless in the face of very large counts of phenomena or data so it makes sense to use a statistical and probabilistic approach. Basically we are saying; this is our best guess! We can be as precise as we need to distinguish probable outcome boundaries but it is still an imprecise fact, not a truth, but as near as we can make it! .
Today people are working on the next best approximations to trochoidal dynamic space.
Here’s a clue: we do not have to describe rotation in just quarter turns . Speed can use any sector of the circle as our rotational magnitude, and indeed roots of unity does that very thing.. My point of view is that pressure expresses itself as a rotational dynamic, that is pressure co ceived as volume energy transforms through a surface and around a surface into curvilinear force vectors. We may depict these by ensembles of tangent vectors or we may resolve this into the motion of a pressure bubble in its environmental space, bearing in mind space is not empty .
That is the magnetic current the leads the way forward!
I often forget about anti matter, but it is evidence that the particle description is flawed. . Rotational dynamics naturally includes such modes of behaviour. Annihilation is not a true depiction of the null rotation. Nodes in 3D are typically of 3 types, positive, negative and neutral, neutral nodes are highly unstable dynamically .
What of non empty space? The magnetic flux in this medium has its viscosity and relaxation time , and this is apparent at high velocities rotational or compressional. Then the elasticity in the medium over short range also becomes apparent . The elasticity coefficient even goes over unity as we approach zero distance of separation . . That means an object can rebound with greater velocity than its incident velocity! What that indicates is that the system is not isolated, energy flows I to the system either by environmental factors or alteration of initial internal states of the objects rotated or compressed in the medium.
On the microscopic scale this is observed as quantum weirdness in the data, when viewed classically. When viewed from the point of view of a Planck quantum, it means that the other factors vary or are varied by our mathematics in order to maintain the quantum as fixed, invariant, and definite. We can not be certain this is the case at these scales so probability descriptions are forced on us, but at larger scales the variations cancel out by interference so that we literally ignore the variation at the boundary of large objects and concentrate on an assumed uniform centre. . That is why Einsteins theories and quantum appear to be in disagreement, and quantum field theory can not give a geometry of them both. It requires a fractal geo entry to describe our non empty space with all our assigned observed behaviours..
We have a Quaternion medium that enables us to model fractal geometric dynamics in a non empty space. .if you visit Fractalforums.com and Fractalforums.org you will gain a sense of how trochoidal dynamics describes forms and dynamics in our observed experience. .
In my thread on Twistor on Fractalforums.com I explore the meaning of a wave deformation in 3 dimensions. We often get mislead by the mathematical resemblance of a 2d graph of the sine function to what looks like a physical wave. But the sine function is unphysical. The rotation that is associated to the continued curve is better but also a special not general rotation. It takes a Fourier series in the exponential form to represent a general 2d wave profile and it takes a Quaternion Fournier series to describe a 3 dimensional travelling wave form . . These representations are little used but represent the next descriptive models of dynamic behaviour.
This book by Sadd gives a readable acoount of wave mechanics in an elastic medium. Of course magnetic behaviour is elastic , with contraction being implosive but repulsion being harmonically damped oscillations in certain conditions.
So before I get into any problems with the quantum description of electromagnetic wave, that was seen in the video above, I want to point out the difficulties of using sine waves to explain diffraction . The problem arises when you look at the reflection through a large gap. From the video you can see that the interference pattern of the so-called sinewaves at a point actually lead to a curve in the phases of the sine waves. So, according to this explanation, we do not get a plane wave coming through a large gap. Another exclamations diffraction only occurs at the edges of the large gap. So the exclamation of why a single select create an interference pattern in this particular video is incorrect.
How do you get a plane wave from an infinite number of point sources? The video and also the common way to deal with this problem is to ignore it. The difficulty is glided over or finessed. In fact the difficulty was addressed by Feynman . If one assumes that the Huygen point. Source is a valid way of arriving at the diffraction patterns and the wave patterns of light, The question is how does light then propagate in a forward manner?
The problem is sold, if one uses Rayleigh waves. Feynman of course use the common notion of a Sine wave and was unable to remove it from the necessary spherical nature of the propagation for. Source. However Rayleigh waves and love waves deal with this as you quite effectively. They propagate a wave in a forward direction and do so by moving the point source in a circular orbit. In fact the way propagates in a spherical orbit and even though it is theoretically proven that radiowave cannot propagate in a horizontal direction to its motion of propagation, this is a mistake due to the over reliance on the mathematics. Rayleigh and love waves propagate in all three dimensions but according to the sine functions of the expression. This sine parts of the solutions are typically ignored as the imaginary part of the calculation. This is because the imaginary part of the calculation is misnamed.
I have dealt with the quarter turn quantity in the calculation is in previous posts, I will explain it again in a later post. The point here is, that the railway of oscillates within its region and when it meets a sweet or a gap in oscillates within the gap until it meets a Boundry line, as it meets the Boundry the edge of the gap, it then causes the diffraction patterns and wave patterns to occur. In the meantime, in open space the really observations create a plane way.
The Rayleigh wave actually travels through the interior of a body in a compressed fall. The amplitude of the Rayleigh wave dissipates and deeper into the body it is travelling through so for depths below the surface level of a material ready wave appears as a compression wave, as the forward compression motion is emphasised over the transverse motion of the Rayleigh wave .
Because of this the rainy wave appears as a compression wave in a medium, and spread out spherically in a compression format. However when the railway wave meet a surface or a change in the surface condition it will form an amplitude or a change in their amplitude of the wave .
Rayleigh waves and love waves away we get circle way was that a harmonic in the dissipation. As the web spreads acts vertically the amount of matter that is being pressurised spreads more Finley, and that’s the height of the wave decreases with the spread of the Rayleigh bubble, or rather the Rayleigh Love wave bubble. . Have a great distance these Rayleigh love wave bubbles spread out into plane waves, which approach the whole or its late, and as they passed through the railway and love motions create the patterns which we called diffraction patterns. The question now is how does dispersion in the prison work?
I start with the reference dipole which has 2 dipoles N and S as centres of virginity. . Between them I place a plane orthogonal to the vector NS that joins the 2 dipole centres . These centres are sources of a rotational curvilinear vector dynamic in 3 dimensions, but are a toroidal form not a sphere.
For simplicity I have placed 3 fixed points in the plane whicact as reference points to measure or trace the flow of the curvilinear vectors.
In the p,and the 3 points A,B,C are not collinear and at the centre of the triangle they form is a point M which lies on the vector NS and which is in dynamic rotation, as are N and S. .
As a convention I extend the vector SN to F and the vector NS to E.
The gyre goes NFAMNFBMNFCMN. The other dipole gyre goes SMAESMCESMBES
These gyres can of course be described with more fixed points in space but this is just to give the simplest description of a dynamic toroidal gyre that will interact trochoidally.
The plane in this scheme ABC has a resultant of xero rotation and zero curvilinear force induction toward M. This represents the Bloch Wall. At the points E and M and F the rotation is maximal and a net through force from E to F is the resultant. When E Nd F are brought together the resultant is a through force induction for any so referenced magnetic dipole, when E andE or F andF are brought together the resultant is a dynamic null which represents an accumulation of through flux or external flux. This accumulation now acts on the poles to spread them apart. ,.
So the magnetic force arises not from an internal source but from a an accumulation of an external pressure source either drawn through the bodies of the magnet forms into the space between or drawn into the space between the magnet forms directly from the environment,
Why did Rayleigh produce the ultraviolet catastrophe?
He forgot about damped oscillation. .
In real space the aether has a damping effect proportional to pressure. Density, viscosity and relaxation time.. so a rotation will not continue forever without some input. . The input is external and or internal to the system, as in a magnetic system. .
In a black body the forcing input is external. So once a cavity is defined the rotations in that cavity will be defined by its pressure density and viscosity and relaxation time characteristics. . For a gas in a container we study this under organ pipe harmonics.
For a closed black body with a tiny outlet the rotational wave that comes out will be diffracted. . So not only will radiation be quantised, it will also be subject to the interference pattern of diffraction. . The intensity distribution would therefore be distributed in a normally distributed bell shaped curve with damped maxima at different angular or radial arc displacements from the direct line of measurement. . We should expect for any fixed cavity that the higher frequencies would be damped because the external input is absorbed by the fundamental frequency of the cavity most effectively.
Remembering that we are dealing with Rayleigh/Love waves not simple sine waves gives us the smooth distribution curves we get from black bodies , especially the sun. In such systems we also have a distribution of open and closed cavities if we accept the sun is a condensed matter body not a gas only,
The magnetic patterning we see on the sun emphasises that this is a magnetic universe with a non empty rotational space dynamic at all scales.
A fractal Magnetic Universe.
In condensed material the viscosity reflects the characteristic of embeddable strain. . Clearly, as phase changes the amount of embeddable strain diminishes , but it is never zero. The result of this observation is that higher frequency oscillations are required to propagate a wave in a gas. .
It should be clear, then that a gas or a plasma is not an aether! The magnetic medium embeds magnetic induction strain. . What that is in terms of anything else is beyond our ability to explain. Feynman could not explain because it is a fundamental. If we accept it as such then we can explain everything else in terms of it. . Thus a wave is the propagation of this embedded strain by stress in the medium, which has a certain relaxation time and a certain viscosity or inertia to strain. . While you may be imagining this in terms of lineal stress it is more general to think of this as rotational stress.. certain strains produce rotational stress that exceed the relaxation limit, and so the material or medium fractures , either explosively or plasticky. . Similarly certain strain will jam stressed materials together plasticky or implosively. .
It is also worth noting that the effect of a Rayleigh/Love wave will extend beyond its surface or impedance boundary into the medium on the other side of the boundary. . Nodes and anti nodes for standing waves define the strain on the other medium.
When someone tells you a travelling wave does not move, do not believe them, it rotates the strain back and forth in a rotational/ pendular motion until this strain returns to equilibrium by dissipation/ radiation of the excess. If the medium itself is moving in block motion this will not effect the wave motion relative to the medium, but of course the Doppler effect results from the medium itself moving non linearly.
Travelling waves move the strain, standing waves conserve the strain unless it dissipates by som other means.
While very easy to follow, certain assumptions are made that mislead. The main one is that for a fixed cavity all the harmonic standing waves have the same amplitude. This is not possible in the string case with both ends as nodes because the string has a certain length and can only be stretched so far. So as the frequency increases the amplitude must necessarily decrease. We see this assumption leads to Rayleigh ultraviolet catastrophe, whereas Planck by concentrating on springs has to include damped oscillations. .
In the same way assuming an orbit for the electron rather than a plum pudding region leads to quantisation. But using a rotational dynamic from the get go also gives quantisation for rotational dynamics. The radiation of the strain by the stresses in the plum pudding analogy are just as quantised as in any other model, because it is rotation that quantised.
Trochoidal dynamics can model the general behaviours of such systems with the proper constraints and boundary conditions.