Copyright(C) 2024,2025 Luke Kenneth Casson Leighton. All Rights Reserved
Contact: lkcl@lkcl.net. Date: 29 Dec 2024. Updated: 13 Jan 2025.
Ref: viXra:2501.0068 v2 DOI:http://dx.doi.org/10.13140/RG.2.2.28905.61285

The de Vries formula: a transcendental solution for Standing Travelling Circular waves

Abstract:

Exploration of the de Vries formula[1] and its accuracy[2] (following removal of SI2019 assumptions) to latest experimental uncertainty[3] has led to a hypothesis that the formula is a transcendental solution of an infinite-summed (non-frictional) circular traveling and standing wave: a solution of harmonic sympathetic resonance.

The de Vries formula is a solution to a more general problem of looped harmonic resonance, where a circle is simply a special case of a loop. Where an open pipe (flute) has a harmonic frequency progression $\lambda/2$ and likewise a closed pipe (organ) $\lambda/4$, the de Vries Formula is instead a solution of sympathetic resonance only in $\lambda$: a circle where the phase-locking is of the travel distance.

In other words: the key is that the (frictionless) wave travels round the loop, and would normally phase-lock with itself. However this produces a travelling wave. If however the wave is slightly longer it becomes a standing travelling wave as well. The interaction between each time round the loop thus results in a Triangular pattern (each contributing term in $\Gamma$) The de Vries Formula, matching this definition, and also being self-referencing, may be expressed as:

$$\alpha = \Gamma^{2} (e^{-\frac{\pi}{4}}){^{2}} ~~~~~ \Gamma = \su... ...\infty} \frac{\alpha^{n}}{\left(2\pi\right)^{T_n}} ~~~~~ T_n = \sum_{k=0}^{n} k$$ (1)

If each travel round the loop is expressed in terms of $e^{-i\theta}$, then each term of the sequence is a standard sum of frequencies:

$$e^{-i({\frac{\alpha}{2\pi}})} ~ e^{-i({\frac{\alpha}{(4\pi)}})} ~... ...{\alpha}{(6\pi)}})} ~ ... ~~~~~ \sum_{k=1}^{k} e^{-i({\frac{\alpha}{(2k\pi)}})}$$ (2)

From the contribution of each repetition to the travelling-standing wave comes naturally the Triangular Number $T_n$, given that frequency multiples of $2\pi$ on a circle are all equivalent to $2\pi$. The concept is better described by figure 2, Chiatti[5].

In summary: $\alpha$ is the solution that provides a standard (infinite progressive) standing wave superposition, but it is a travelling one (figure 4, Riggs[4]) with the wave phase-locked with itself to the length of the circumference. However when that same travelling wave continues its journey (infinitely, assuming a frictionless system) it is $\Gamma$ that brings in the small corrective factor to make this travelling wave also a standing wave with all of its infinitely-travelling recurrences of itself.

References

1

Hans de Vries, An exact formula for the Electro Magnetic coupling constant.

http://chip-architect.com/news/2004_10_04_The_Electro_Magnetic_coupling_constant.html

October 4, 2004.

2

Luke Kenneth Casson Leighton, Implementation of de Vries formula for alpha in python3,

http://lkcl.net/reports/fine_structure_constant/alpha.py.

3

Léo Morel, Zhibin Yao, Pierre Cladé, Saïda Guellati-Khélifa, Determination of the fine-structure constant with an accuracy of 81 parts per trillion, https://www.nature.com/articles/s41586-020-2964-7

4

Peter J. Riggs, Revisiting Standing Waves on a Circular Path,

http://hdl.handle.net/1885/287028, DOI:https://doi.org/10.1119/10.0003461

5

Leonardo Chiatti, Quantum Jumps and Electrodynamical Description,

https://www.researchgate.net/publication/320110017, International Journal of Quantum Foundations 3 (2017) 100 - 118

About this document ...

The de Vries formula: a transcendental solution for Standing Travelling Circular waves

This document was generated using the LaTeX2HTML translator Version 2021 (Released January 1, 2021)

The command line arguments were:
latex2html -split 0 -no_navigation travelling_standing_circular.tex

The translation was initiated on 2025-01-13