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 β-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).

Keywords: Gerstner waves, Lagrangian coordinates, vorticity, Cauchy invariants, edge waves, Ptolemaic flows, rotating fluid, f-plane approximation, equatorially trapped waves PACS: 47.35.Bb DOI: 10.3367/UFNe.2021.05.038980 URL: https://ufn.ru/en/articles/2022/5/c/ 001112520100003 2-s2.0-85160618208 2022PhyU...65..453A Citation: Abrashkin A A, Pelinovsky E N "Gerstner waves and their generalizations in hydrodynamics and geophysics" Phys. Usp. 65 453–467 (2022) BibTexBibNote ® (generic)BibNote ® (RIS)MedlineRefWorks Received: 1st, February 2021, revised: 9th, April 2021, accepted: 3rd, May 2021 Оригинал: Абрашкин А А, Пелиновский Е Н «Волны Герстнера и их обобщения в гидродинамике и геофизике» УФН 192 491–506 (2022); DOI: 10.3367/UFNr.2021.05.038980