Gerstner waves and their generalizations in hydrodynamics and geophysics

To mark 220 years since the appearance of Gerstner's paper that proposed an exact solution to the hydrodynamic equations, an overview of exact solutions for water waves is given, each of which is a generalization of the Gerstner wave. Additional factors are coastal geometry, fluid rotation, varying pressure on the free surface, stratification, fluid compressibility, and background flows. Waves on a rotating Earth are studied in the $f$-plane approximation, and, in the near-equatorial region, also in the $\beta $-plane approximation. The flows are described in Lagrangian variables. For all waves in the absence of background flows, the trajectories of liquid particles are circles, as in the Gerstner wave (hence, their common name—Gerstner-like).