Standing wave theory

Figure 2-22. McLachlan s selection [13] of Bentley s snowflake images [14] to illustrate the coordinated growth of the six branches of a snowflake based on the standing wave theory.

Whereas the profile in linear wave equations is usually arbitrary it is important to note that a nonlinear equation will normally describe a restricted class of profiles which ensure persistence of solitons as t — oo. Any theory of ordered structures starts from the assumption that there exist localized states of nonlinear fields and that these states are stable and robust. A one-dimensional soliton is an example of such a stable structure. Rather than identify elementary particles with simple wave packets, a much better assumption is therefore to regard them as solitons. Although no general formulations of stable two or higher dimensional soliton solutions in non-linear field models are known at present, the conceptual construct is sufficiently well founded to anticipate the future development of standing-wave soliton models of elementary particles. 

Whittaker s early work [27,28] is the precursor [4] to twistor theory and is well developed. Whittaker showed that a scalar potential satisfying the Laplace and d Alembert equations is structured in the vacuum, and can be expanded in terms of plane waves. This means that in the vacuum, there are both propagating and standing waves, and electromagnetic waves are not necessarily transverse. In this section, a straightforward application of Whittaker s work is reviewed, leading to the feasibility of interferometry between scalar potentials in the vacuum, and to a trouble-free method of canonical quantization. 

This means that the special relativity theory does not predict any shift of interference fringes that is contrary to the experiments. The standing-wave approach of Sagnac-type experiments allows a freedom in the definition of c+ and c instead of Eqs. (9) and (11), but the second postulate of the special relativity theory is out of this range. 

In the spirit of the standing-wave picture of Sagnac-type experiments, this theory needs to recalculate the result of the Michelson-Morley experiment as well. In the M-M experiment there is a new unknown hidden parameter cp, which denotes the speed of light in the direction perpendicular to the earth s velocity. The traveled path of light in the perpendicular arm Xp = 2Tcp [dim X = meter], [where cp is speed of light perpendicular to the velocity... 

We now check whether Eq. (1), with S /3 = e2/ and modified as above to account for finite propagation time, can explain our data. The unknown parameters are the resistance Rq and the effective environment noise temperature Tq. We checked that the impedance of the samples was frequency independent up to 1.2 GHz within 5%. Fig. 2 shows the best fits to the theory, Eq. (1), for all our data. The four curves lead to Ro = 42 12, in agreement with the fact that the electromagnetic environment (amplifier, bias tee, coaxial cable, sample holder) was identical for the two samples. We have measured the impedance Zenv seen by the sample. Due to impedance mismatch between the amplifier and the cable, there are standing waves along the cable. This causes Zenv to be complex with a phase that varies with frequency. We measured that the modulus Zenv varies between 30 12 and 70 12 within the detection bandwidth, in reasonable agreement with f o = 42 12 extracted from the fits. 

A little-known paper of fundamental importance to modern atomic theory was published by Hantaro Nagaoka in 1904 [10]. Apart from oblique citation, it was soon buried and forgotten. With hindsight it deserved better than that. It contained the seminal ideas underlying the nuclear model of the atom, the standing-wave nature of orbital electrons and radiationless stationary states. It was so far ahead of contemporary thinking that later imitators either failed to appreciate its significance, or pretended to be unaware of it. 

In terms of Wheeler-Feynman absorber theory [47] a standing wave, consisting of synchronized retarded and advanced components, originating at emitter and absorber respectively, constitutes the wave function of a photon and it extends over the entire region between emitter and absorber. 

Debye Theory of the Heat Capacity of Solids. Debye assumed that a cubic crystal of side L and volume V = Lr can be taken as a vacuum (German Hohlraum) that supports a set of standing waves, each with form... 

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