  The mechanics of wave motion in a medium are founded in conservation laws for
the physical quantities that the waves carry, combined with the constitutive
laws of the medium, and define Lorentzian structures only in degenerate cases
of the dispersion laws that follow from the field equations. It is suggested
that the transition from wave motion to point motion is best factored into an
intermediate step of extended matter motion, which then makes the
dimension-codimension duality of waves and trajectories a natural consequence
of the bicharacteristic (geodesic) foliation associated with the dispersion
law. This process is illustrated in the conventional case of quadratic
dispersion, as well as quartic ones, which include the Heisenberg-Euler
dispersion law. It is suggested that the contributions to geodesic motion from
the non-quadratic nature of a dispersion law might represent another source of
quantum fluctuations about classical extremals, in addition to the diffraction
effects that are left out by the geometrical optics approximation.
