Shallow-water wave refraction as geodesic motion in an optical metric

Refractionofshallow-waterwavesovervariablebathymetryistypicallytreatedinoceanog-

raphy via the continuous Snell law or by numerical ray tracing. We show that the same

physics admits a compact differential-geometric formulation: rays are geodesics of a two-

dimensional optical metric. Starting from Fermat’s principle with n = 1/√gh, we derive

the explicit Christoffel symbols, recover the Snell invariant, obtain a closed-form ray tra-

jectory on a plane beach, and compute the Gaussian curvature of the optical metric, which

controlsfocusingthroughthegeodesicdeviationequation. Whilestandardinopticsandrel-

ativity, this formalism is rarely used in coastal wave theory and offers a unifying theoretical

language.