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.
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