Abstract
We present some explicit solutions (given in Eulerian coordinates) to the three-dimensional nonlinear water wave problem. The velocity field of some of the solutions exhibits a non-constant vorticity vector. An added bonus of the solutions we find is the possibility of incorporating a variable (in time and space) surface pressure which has a radial structure. A special type of radial structure of the surface pressure (of exponential type) is one of the features displayed by hurricanes, cf. Overland (Earle, Malahoff (eds) Overland in ocean wave climate, Plenum Pub. Corp., New York, 1979).
Highlights
The ubiquitous manifestation of water wave propagation has been given mathematical attention as early as the eighteenth century through the works of Bernoulli, Euler, Lagrange and d’Alembert
One drawback is the pronounced shortage of explicit solutions to the governing equations of water flows with a non-flat free surface
In the absence of explicit solutions, most exact solutions pertaining to surface water wave propagation were obtained by perturbation approaches employed to replace the nonlinear governing equations by approximate models which constituted the basis for an assortment of theoretical studies
Summary
The ubiquitous manifestation of water wave propagation has been given mathematical attention as early as the eighteenth century through the works of Bernoulli, Euler, Lagrange and d’Alembert. While the Lagrangian perspective conveys important insights into the flow evolution by tracking down the path of particular particles, it is desirable to know the velocity field, the pressure function as well as the shape of the free surface at any given time instant and physical location, prospect known as the Eulerian picture. Meeting the latter demand are the studies of Constantin and Johnson [10,12]
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