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Waves

So far EthnoPhysics has considered a generic particle P by objectifying some repetitive chain of events written as

\Psi^{\mathsf{P}} = \left( \mathsf{\Omega}_{1}, \, \mathsf{\Omega}_{2}, \, \mathsf{\Omega}_{3} \, \ldots \,          \right)

Each cycle  \mathsf{\Omega} is defined by sensation. And they must be repetitive so that P can be recognized, so  \mathsf{\Omega}_{1} = \mathsf{\Omega}_{2} = \mathsf{\Omega}_{3} \, \ldots and so on. But we might just as well understand such a recurring sequence of sensations to be a wave train. That is,  \Psi could represent some sort of of periodically undulating or fluctuating perception. This interpretive ambivalence is called wave-particle duality and it has been contentious during the development of physics. However for EthnoPhysics there is no quandary; scientific facts and theories are founded on sensation, and whether we call these perceptions particles or waves is just a question of convenience. If feelings are localized, then we talk about particles. Or if sensory phenomena seem to have some extended quality, then we often use words like wave, wavenumber, wavelength, etc. In between, we might speak of particles that are in excited states. These terms are developed from a discussion of quarks as follows. Let each cycle of P be a bundle of  N quarks written as

\mathsf{\Omega} = \left( \mathsf{q}^{1}, \, \mathsf{q}^{2}, \,  \mathsf{q}^{3} \, \ldots \, \mathsf{q}^{i} \, \ldots \, \mathsf{q}^{N} \right)

And let each quark be described by its phase  \delta_{\theta} along with its radius vector  \overline{\rho}. Then the wavevector of P is defined as

\displaystyle \overline{\kappa}  \equiv \left( \dfrac{1}{\rho_{in}^{2}} - \dfrac{1}{\rho_{out}^{2}} \right) \sum_{i=1} ^{N}   \delta_{\theta}^{\, i}   \;  \overline{\rho}^{i}

where  \rho_{in} is the inner radius and  \rho_{out} is the outer radius of P. Substituting-in the definitions for these radii gives the wavevector in terms of the coefficients of down quarks as

\displaystyle \overline{\kappa}  =  \frac{  k_{\mathsf{F}}}{hc}  \left[ \left( \frac{8}{ \Delta n^{\mathsf{D}} }  \right)^{\! 2}       -       \left(  \frac{8}{ N^{\mathsf{D}} } \right)^{\! 2}   \right] \cdot  \sum_{i=1}^{N}   \delta_{\theta}^{\, i}   \;  \overline{\rho}^{i}

But for a perfectly free particle,  \left| \Delta n^{\mathsf{D}} \right| \gtrsim 8 and  N^{\mathsf{D}} \to \infty. So the wavevector of a free particle can be put plainly as

\displaystyle \overline{\kappa} = \frac{ k_{\mathsf{F}} }{hc} \sum_{i=1}^{N} \delta_{\theta}^{\, i}   \;  \overline{\rho}^{i}

Note that \overline{\kappa} is relative characteristic because the phase depends on the juxtaposition of P with some frame of reference. Let this frame be steady so that quarks do not change phase. Then if some free particles interact like \mathbb{X} + \mathbb{Y} \leftrightarrow \mathbb{Z} , then their wavevectors will combine as

\overline{\kappa}^{ \mathbb{X}} + \overline{\kappa}^{\mathbb{Y}}  = \overline{\kappa}^{\mathbb{Z}}

This is because quarks are conserved and  \overline{\rho} is defined from sums of quarks. The radius vector also relates particles and anti-particles by \overline{\rho} \mathsf{(P)} = - \overline{\rho} \mathsf{(\overline{P})}. So if quarks are swapped with anti-quarks, without altering the phase, then

\overline{\kappa} \left( \mathsf{P} \right) = - \,  \overline{\kappa}  \left( \overline{\mathsf{P}} \right)

The average wavevector describes some hypothetical typical quark in P using the ratios \widetilde{\kappa} \equiv \overline{\kappa} / N where  N is the total number of quarks in P.

Inertial Frames of Reference

Let some well-known particle F be employed as a frame of reference. This just means that we use F to describe changing phenomena. Otherwise a reference frame is a compound quark like any other particle, so it can be characterized by its quark coefficients and wavevector \overline{\kappa}. For example, a rigid frame of reference always has the same radius vector. Another important special case is when each component of the average wavevector of F is zero

\widetilde{\kappa} \equiv \dfrac{ \overline{\kappa} }{N} = \left( 0, 0, 0 \right)

then we say that F provides a perfectly inertial frame of reference. This condition is approximated when the total number of quarks  N is enormous. Because if the total number of quarks is huge, there will likely be some mix of quarks and anti-quarks making the  \Delta n terms in the radii of the numerator tend toward zero, even as the denominator gets larger. If a frame of reference is rigid, inertial and non-rotating then we call it an ideal reference frame.

Waves and Wavelengths

A wavenumber can be specified by the norm of a wavevector. It is written without an overline as   \kappa \equiv \left\| \, \overline{\kappa} \, \right\|. Then the wavelength of P is defined as

\lambda \equiv \begin{cases} \hspace{15 px} 0 \; &\mathsf{\text{if}} \; \kappa =0 \\ \; 2\pi / \kappa    \; &\sf{\text{if}} \; \kappa \ne 0 \end{cases}

Thus \lambda takes a logically required, discontinuous jump when \kappa =0. And it leaps to zero, so perhaps we could say it collapses. But keep in mind that this fracture is just one of many. The wavelength is discontinuous for all values because it is defined from quark coefficients, and quark coefficients are always integers.

XXX

Sensory Interpretation: Radius vectors are defined from dynamic quarks, not baryonic quarks. So the wavevector, wavenumber and wavelength can only represent somatic and visual sensations, not thermal perceptions. And, if a frame of reference is inertial, it is big and grey. Next we have more to say about excited particles.

Waves of repetitive sensation are graphically displayed in this rattan weaving from Indonesia.
Ajat basket, Penan people. Borneo 20th century, 20 (cm) diameter by 35 (cm) height. Hornbill motif. Photograph by D Dunlop.