Narrow Results

The water wave transmission properties and the frequency spectra of one-dimensional combination bottom of water and mercury with different filling fractions are studied by the transfer matrix method. For the periodic bottoms (PBs), the effect of the steps' numbers and the width on the band gaps are discussed. Each of whole band gaps is the juxtaposition of the gaps of three kinds of PBs, without covering. The numerical results show that the band gaps could be enlarged effectively by choosing the steps' width properly.

A highly efficient and accurate numerical scheme for initial and boundary value problems of a two-dimensional Boussinesq system which describes three-dimensional water waves is used to study in details the oblique interaction of a solitary wave and the evolution of solitary waves coming out of a narrower channel.

In this article we explain the well-posedness of the initial value problem for water waves, and the shallow water and long wave approximations for water waves. Especially, we explain that the solution of the full water wave problem can be approximated by the solution of the shallow water equations, the Green–Naghdi equations, the KdV equation, the Kawahara equation, the forced KdV equation, and the Benjamin–Ono equation in some scaling regimes.