We derive a depth-averaged equation for the magnetic field induced by long surface gravity waves over variable seabed. The equation is verified using known analytical results and a novel numerical model for magnetic anomalies over variable bathymetry. Unlike amplitude-based theories, our results show that the magnetic response is governed by the forward energy flux associated with the surface gravity wave. This reframes the physics of long-wave magnetics and provides a new basis for interpreting geomagnetic observations.
Renzi et al. (2026) studied this question.