Inertio-gravity Poincaré waves and the quantum relativistic Klein-Gordon equation, near-inertial waves and the non-relativistic Schrödinger equation

Shallow water inertio-gravity Poincaré waves in a rotating frame satisfy the Klein-Gordon equation, originally derived for relativistic, spinless quantum particles. Here, we compare these two superficially unrelated phenomena, suggesting a reason for them sharing the same equation. We discuss their energy conservation laws and the equivalency between the non-relativistic limit of the Klein-Gordon equation, yielding the Schrödinger equation, and the near-inertial wave limit in the shallow water system.