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Abstract

Nonlinear interactions between sea waves and the bottom are a main mechanism of energy transfer between the different wave frequencies in the near-shore region. In this region, nonlinear interactions act much faster than in deep water due to quadratic resonance interactions. One of the methods for solving this flow regime is using quadratic nonlinear mild-slope (MS) type wave models. These models consist of a linear mild-slope type equation for each wave harmonic coupled by quadratic nonlinear terms to all other harmonics.

The first part of the talk will discuss the various options for formulating the magnitude of the wave number rather than the commonly used heuristic choice. This allows constructing models that allow for different types of solution methods, and gives a better overview for extending the formulation to two-dimensions.

The second part will discuss the phase functions and the directions of the wavenumber vectors. This information is needed for constructing this type of models, and the problem of its formulation is what limits these models to one-dimensional propagation, or to two-dimensional ones with some crude assumptions.

In the present work, governing equations for the wavenumber vectors and the phase functions are constructed in order to allow for rigorous derivations of each type of solution method for various wave propagation characteristics. This allows constructing equations for the two-dimensional propagation of oblique incident waves in various angles that interact both with each other and with the seabed. A perturbation approach is used in order to simplify these equations while keeping superior accuracy with respect to other models.

Another extension to the commonly used models that will be presented, is the inclusion of nearly resonant interactions. For oblique propagation toward a beach with parallel bathymetry lines, this inclusion allows constructing a higher order correction that changes the nature of the solution causing the waves to evolve also in the lateral direction.

In order to address as well people that are not from the field of water waves, some basic concepts of wave propagation will be discussed, and the main mechanisms for nonlinear energy transfer will be explained in an intuitive manner.

"The asymptotic structure of three-dimensional gravity with a negative cosmological constant is reviewed with a special emphasis on the central charge that appears in the asymptotic conformal symmetry algebra. Higher spin extensions, which were recently investigated, are discussed"