Stable Three-Dimensional Biperiodic Waves in Shallow Water

摘要

Waves in shallow water are inherently three-dimensional and nonlinear. Experiments are presented herein which demonstrate the existence of a new class of long water waves which are genuinely three-dimensional, nonlinear, and of (quasi-) permanent form. These waves are referred to as biperiodic in that they have two real periods, both temporally and spatially. The waves are produced in the laboratory by the simultaneous generation of two cnoidal wave trains which intersect at angles to one another. The resulting surface pattern is represented by a tiling of hexagonal patterns, each of which is bounded by wave crests of spatially variable amplitude. Experiments are conducted over a wide range of generation parameters in order to fully document the waves in the vertical and two horizontal directions. The hexagonal-shaped waves are remarkably robust, retaining their integrity for maximum wave heights up to and including breaking and for widely varying horizontal length scales. The Kadomtsev-Petviashvili equation is tested as a model for these biperiodic waves. This equation is the direct three-dimensional generalization of the famous Korteweg-deVries equation for weakly nonlinear waves in two dimensions.