  The possible relation of the wave nature of particles to gravitation as an
emergent phenomenon is addressed. Hypothetical particles are considered as
spatially confined oscillations (SCOs) and are constructed through the
superposition of plane waves. The effect of a continuously changing refractive
index (speed of propagation field) on SCOs is calculated and the continuous
Ibn-Sahl--Snell law of refraction is derived. Refracted plane wave constituents
of SCOs in an inhomogeneous medium cause the oscillation as a whole to
accelerate as an entity. This acceleration is described by a geodesic equation,
in much the same way as in general relativity. The proper time of an SCO can be
defined via its oscillation frequency. The proper time and its change along the
trajectory are equivalent to a particle in general relativity as described by
the 0th component of its geodesic equation. An SCO in an inhomogeneous
refractive index field exhibits general relativistic properties based on basic
wave mechanics. Properties derived from direct calculations are length
contraction, gravitational red- and blueshift and Thomas precession. The
presented theory is an approximation for oscillations which are small compared
to changes in the refractive index field. SCOs in an inhomogeneous medium may
thus yield a naturally emerging particle-field interaction with general
relativistic properties and may allow a useful vantage point on the nature of
gravitation using classical-wave experiments.
