A modern look at a 200-year-old wave

The motion of waves on the surface of the ocean has motivated and perplexed physicists for centuries. Great minds, such as Euler, Stokes, and Airy, have puzzled over the equations to describe wave motion and their resulting solutions. But waves are difficult to represent analytically, and many of the well-known attempts make simplifying assumptions that limit their applicability to real-world situations.

Anatoly Alexandrovich Abrashkin formulated a solution for a non-stationary vortex gravity-capillary wave in deep water. In contrast to stationary waves, also known as Gerstner waves, this solution incorporates movement of the wave’s free surface, making it more useful for practical applications.

The traditional Gerstner wave, formulated by Franz Josef Gerstner in 1804, features particles that move in circles with the radius dependent on their depth. However, in this formulation the pressure on the surface is constant, so Gerstner waves cannot account for external forces like wind.

Abrashkin’s solution allows for pressure variations and accounts for the water’s surface tension, reproducing several characteristics of real-world waves, including rogue waves and breaking waves.

“The hydrodynamic equations are written for complex coordinates of the fluid particle’s trajectory, which depend on complex Lagrangian variables,” said Abrashkin. “This approach allows us to find an exact solution for describing unsteady vortex flows, depending on two arbitrary analytical functions and two time constants.”

Abrashkin sees his solution becoming a starting point for more detailed investigations of fluid flows.

“Despite the more complex form of the Lagrangian hydrodynamic equations, the free-surface boundary condition is an advantage and can become a methodological basis for constructing theories of wave-wind interaction,” said Abrashkin.

Source: “Gerstner gravity-capillary waves and their collapse mode,” by Anatoly Alexandrovich Abrashkin, Physics of Fluids (2026). The article can be accessed at https://doi.org/10.1063/5.0343860 .