ON THE EXTREME POSITION OF PROPAGATION OF BICHROMATIC WAVE IN HYDRODYNAMIC LABORATORY

Abstract

This paper concerns with the down-stream propagation of waves over initially still water. Such 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 it 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. The model is used in this paper is a Korteweg-de Vries (KdV) model and third order approximation theory to calculate the approximate extreme positions for a class of wave. This class is the wave-groups that originate from initially bi-chromatics type of wave, described by superposition of two monochromatic waves. It is shown that, the extreme position depends on the amplitude and the wave length of the group. The theoretical results are verified with numerical as well as experimental results for comparison.