Dynamic properties of almost monochromatic standing waves

Abstract

For a scalar field propagating at light velocity c, we show that kinematic and dynamic properties of almost monochromatic standing waves, with frequency Ω(x, t)=ω±δΩ(x, t), where ω is constant, and δΩ(x, t)�ω, are formally identical with mechanical properties of matter. They are both described by equations with the same mathematical structure. The energy conservation stems from stability in time, while the variational principle stems from stability in space. In classical mechanics of a particle, the relativist equations correspond to the geometrical optics approximation as ω→∞. The quantum mechanical equations correspond to the wave optics approximation, in which wave homogeneous Fourier relations are replaced by the material Heisenberg relations.