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Harmonics Theory - The Physics and Maths （第二部分）
| Speed of light |
In general relativity, the speed of light is considered constant, but the amountof space in a region may be different from what one expects, somewhat in themanner of Dr Who's telephone box, with more space on the inside than there really ought to be. However, as mentioned earlier this same explanation is notused for the refraction of light in glass where it is quite happily admitted that the speed of light varies rather than saying that the glass is bigger onthe inside than the outside.
It is possible to rephrase general relativity so that space is euclidean and the velocity of light varies. Such an approach is much easier on the mind as humans seem to be almost preprogrammed with the concepts of euclidean space.
When the wave equation was discussed comparisons were made to guitars and drums. In general, wave equations have the property that something is distorted and then"tries" to return to its original shape and in the process overshoots and ends up vibrating for a period. Clearly the possibilities in three dimensional space are much greater than in a one dimensional guitar string or two dimensional drum.
Even in these lesser dimensions, if the plucking or beating is very hard then that action may actually tighten the string or membrane, resulting in more tension and a slightly faster wave velocity. Such action will cause an instrument to be slightly higher in note although as the vibration dies down it will return to normal. Understanding that matter is very concentrated electromagnetic energy and that massive objects have great gravity, it is reasonable to also expect the presence of such "things" will cause a change in light velocity consistent with the alternative interpretation of general relativity and the behavior of light in matter.
This variation of light wave speed with changes in the tension of the electromagnetic medium due to high concentrations of energy is the non-linearity that was mentioned and concludes the preparations for understanding a whole series of behaviors of the universe that were not previously realized.
Actually it is not necessary for physicists to change their view as suggested above in order to understand what follows, but it does make things easier on the mind.It is accepted that Einstein's general relativity does make Maxwell's equations into a non-linear system. That is sufficient.
| Non-linear Waves make Harmonics |
In any non-linear system waves will develop harmonics. This means that wave shapes gradually change over time. An example of this is when ocean waves approach the shore the wave velocity changes with changing depth and so the peaks of waves begin to catch up to the troughs, eventually breaking.
Consider a standing wave that fills the entire universe or, if you think the universe is infinite, then consider a large wave that satisfies some characteristic conditions of the universe. If the wave equation for the universe were linear,then such a wave would carry on wobbling in place for ever, never interacting with itself. However as the wave equation is non-linear the wave must change shape over time. For example, this changing in shape might be to have a sharper peak and flatter trough.
In presently accepted cosmology there may not be sufficient time in the universe for very large waves to develop in shape due to non-linearities. However the indication is that the measured parameters of such models show an acceleration parameter, which is another way of saying that the past may have been much longer than the initial indications. For the purpose of what follows the assumption will be made that there is no difficulty with large universal waves existing for very many cycles, which means that very particular values of cosmological parameters are necessary.
When a wave has a shape different from a sine wave then it is said that harmonics are present. Real musical instruments also have this property as when we pluck aguitar string it makes waves that fit not just one wave in the string, but also other waves that fit two, three, four and more waves exactly into the string.These other waves will always divide the original wave by an integer, and inthe linear situation will have frequencies which are exact multiples of the fundamental wave.
The three dimensional universe is no different in this respect, the fundamental wave will have harmonics which are exact divisions of the wave length, and if the non-linearity is not too great, multiples of the original frequency.However something additional happens in three dimensional standing waves that does not happen in the one dimensional case of the guitar. By the interaction of the three dimensions, these new waves also become standing waves and through the non-linearity also produce harmonics. This production of further harmonics by the harmonics is something that has not previously been investigated. Be ready for some entirely new understanding of the nature of waves in the universe.
| Many Harmonics |
Based on the present understanding of physics the universe may support large standing waves and because of the non-linearity of the wave equation we know that every wave will produce harmonics and in three dimensions these waves will also be standing waves. Taking a brave step from the correct understanding of the work of Wheeler and Feynman the assumption is that there is nothing but these waves.So forgetting all other known physics we state a single principle:
| The Universe consists of a standing wave which develops harmonically related standing waves and each of these does the same. |
From this simple principle the pattern of frequencies of the waves formed in the universe can be calculated. For the following calculations the assumption is that there initially exists a single frequency of a universal wave and that frequency will be called 1. We do not need to know 1 what to do the calculations, but it might be 1 cycle per period of about 14 billion years if that helps people to have some idea in their minds. When frequencies of 2 and 3 and other numbers are mentioned that will mean 2 or 3 or more cycles in 14 billion years. Here is a first look at this process of harmonic generation.
| 1 --> 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 ... 2 --> 4 6 8 10 12 14 16 3 --> 6 9 12 15 4 --> 8 12 16 5 --> 10 15 6 --> 12 7 --> 14 8 --> 16 ...etc |
This table shows that some harmonics are formed in multiple different ways, while others are formed in only one way. We may look at the 12th harmonic compared tothe 11th and 13th. If a harmonic is formed in more ways then, assuming that the energy from each path is equal, it should have more energy in that wave. The situation is actually more extreme than it at first appears, because some of the harmonics going into the 12th harmonic already occurred in multiple ways and so each supplied multiple doses of energy. Altogether the 12th harmonic may be produced in 8 ways which are 12, 6x2, 2x6, 4x3, 3x4, 2x2x3, 2x3x2, 3x2x2.Each of these ways may supply a flow of energy equal to the 11th or 13th harmonic on its own.
To make this strictly true it is necessary that every path is equal. That will be so when the energy flow from any wave to its harmonics is proportional to the energy of the wave itself and proportional to some inverse power of that harmonic number as well. It doesn't matter which inverse power, the consequences are equivalent. This inverse power means that any harmonic mainly loses energy to its second harmonic with less to the third and rapidly diminishing amounts to the higher order harmonics. Such conditions are quite reasonable for a non-linear wave when the non-linearity is not very great.However under very extreme conditions of energy concentration there may need to be some additional elements added to this calculation.
This second look at harmonics generation keeps count of the number of ways in which each harmonic is formed. So as the ways in which the 4th harmonic is formed is found in the totals at the bottom of the table to be 2, that is fed back in at the side to its production of the 8th, 12th and 16th harmonics and so on.
| 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 ... 1 --> 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ... 2 --> 1 1 1 1 1 1 1 3 --> 1 1 1 1 4 --> 2 2 2 5 --> 1 1 6 --> 3 7 --> 1 8 --> 4 ... etc Total 1 1 2 1 3 1 4 2 3 1 8 1 3 3 8 ... |
This little table shows the method of calculating how many ways each harmonic can be formed. Once the method is understood, computers can be used to do the same calculations for an extended range of numbers including very large harmonic numbers. Such calculations have been performed for numbers with 50 digits and more, but there is a practical limit because the number of calculations gets very great.
2019年9月18日下午三点焦炭主力合约J2001分析结果：
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