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
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