  Regardless of number, standing wave modes are by definition noninteracting,
and therefore cannot thermalize by themselves. Doppler shifts due to thermal
motions of cavity walls provide necessary mixing, but also preserve the
amplitudes and phases. The lambda/2 intervals of the modes thus preserved must
have equal energy expectations, say <E>, in the resulting equilibrium. By
definition again, they can be exchanged between modes only in whole numbers and
hence only between harmonics. Each family of harmonic modes is thus
self-contained and is disjoint from other families in such exchanges, and
further, can have no more than one mode excited at any instant. The second
property identifies harmonic families of standing wave modes as the harmonic
oscillators of Planck's theory, since a family can only bear energy equal to
exactly one of E, 2E, 3E, etc These two properties further imply that the
energy expectation gets averaged only over an individual family, as the
equilibrium energy <E(f)> steadily available at a given mode would have
contributions from its entire harmonic family. Planck's equations reemerge, and
radiation quantization arises as a classical rule <E(f)> = <E>f, as a mode must
contain a whole number of exchangeable lambda/2 intervals, but it only concerns
equilibrium states. The result makes Planck constant h an analogue of
Boltzmann's constant k for the frequency domain, and points to postulate-free
explanations of all aspects of quantum and kinetic theories.
