On the maximal temporal amplitude of down stream running nonlinear water waves

Abstract

This paper concerns with the down-stream propagation of waves over initially still water. Such a study is relevant to generate waves of large amplitude in wave tanks of a hydrodynamic laboratory. Input in the form of a time signal is provided at the wave-maker located at one side of the wave tank; the resulting wave then propagates over initially still water towards the beach at the other side of the tank. Experiments show that nonlinear effects will deform the wave and may lead to large waves with wave heights larger than twice the original input; the deformations may show itself as peaking and splitting. It is of direct scientific interest to understand and quantify the nonlinear distortion; it is also of much practical interest to know at which location in the wave tank, the extreme position, the waves will achieve their maximum amplitude and to know the amplitude amplification factor. To investigate this, a previously introduced concept called Maximal Temporal Amplitude (MTA) is used: at each location the maximum over time of the wave elevation. An explicit expression of the MTA cannot be found in general from the governing equations and generating signal. In this paper we will use a Korteweg - de Vries (KdV) model and third order approximation theory to calculate the approximate extreme positions for two classes of waves. The classes are the wave-groups that originate from initially bi-chromatics and Benjamin-Feir (BF) type of waves, described by superposition of two or three monochromatic waves. We show that for initially bi-chromatics signals, the extreme position does not depend on the phases of the mono-chromatic components. For BF signals, however, the phases of the mono-chromatic components influence the extreme position essentially. The theoretical results are verified for the case of bi-chromatics with numerical as well as experimental results; for BF signals we use an analytical solution called the Soliton on Finite Background (SFB) for comparison.