Dimension-Reduced Model for Deep-Water Waves

Michael Bestehorn, Peder A. Tyvand, Thomas Michelitsch
2019 Journal of Applied Mathematics and Physics  
Starting from the 2D Euler equations for an incompressible potential flow, a dimension-reduced model describing deep-water surface waves is derived. Similar to the Shallow-Water case, the z-dependence of the dependent variables is found explicitly from the Laplace equation and a set of two onedimensional equations in x for the surface velocity and the surface elevation remains. The model is nonlocal and can be formulated in conservative form, describing waves over an infinitely deep layer.
more » ... ly deep layer. Finally, numerical solutions are presented for several initial conditions. The side-band instability of Stokes waves and stable envelope solitons are obtained in agreement with other work. The conservation of the total energy is checked. How to cite this paper: Bestehorn, M., Tyvand, P.A. and Michelitsch, T. (2019) Dimension-Reduced Model for Deep-Water Waves. Journal of Applied Mathematics and Physics, 7, 72-92. where d is the layer's depth and a typical lateral length scale (e.g. wave length). A similar expansion can be done for highly viscous problems and the so-called Thin-Film equation is obtained which describes the dynamical behavior of the shape of the film's surface [2] [3] [4] . For convection problems, a projection onto certain z-dependent modes allows for the reduction to 2D systems
doi:10.4236/jamp.2019.71007 fatcat:tvag56l3xzee7fnqd5ifheqrqe