Water Waves: Nonlinear Theory

TL;DR

This paper introduces a new nonlinear mathematical theory for deep water surface gravity waves that overcomes limitations of classical models, ensuring unique solutions with finite energy and revealing previously unknown wave evolution patterns.

Contribution

The novel theory allows for analysis without restrictive assumptions, guarantees solution uniqueness, and describes new wave patterns confirmed by experimental data.

Findings

Identifies new wave evolution patterns

Ensures solution uniqueness without radiation condition

Solutions are valid from initial time to infinity

Abstract

A novel mathematical nonlinear theory of surface gravity waves in deep water is presented, in which analytical analysis of the classical nonlinear equations of fluid dynamics is performed under less restrictive assumptions than those applied by existing theories. In particular, the new theory ensures uniqueness of the solution without the need to employ the so-called radiation condition, and its solutions are such that the liquid always remains at rest at infinity, and the energy supplied to the water by a source of disturbances is finite at all times - all that in contrast with conventional approaches that operate in terms of spatially-infinite harmonic waves. The new theory accounts for the non-linearity of the problem, and yields solutions valid at all times, from zero (the time of setting initial conditions) to infinity. The author describes previously unknown patterns in wave evolution, and confirms those patterns with the experimental results by other researchers.