Waveform Evolution Of "Running" Faraday Waves In A Narrow Tank

Faraday wave has always been a hot topic in the field of fluid mechanics。Theoretically,its generation mechanism has a very clear physical picture of parametric resonance,and experimentally,the experimental conditions required to generate Faraday waves are relatively simple.However,the dynamic behavior of Faraday waves is very complex,and many interesting phenomena can be generated under nonlinear effects.Faraday wave is one of the important systems in the study of nonlinear dynamics.One of the key point in Faraday waves is the discovery and understanding of new experimental phenomena.In this thesis,a special Faraday wave in a narrow tank is introduced and analyzed in detail.From the standing wave excited by the parametric resonance,the waveform is similar to a pair of soliton-like waves is developed.This pair of soliton-like waves runs back and forth along the tank with a fixed period,and the wave still maintains such a steady motion after a long time evolution.The pair of the soliton-like waves do not fade away over time or destabilize the system.The first and second chapters mainly introduce the research background and the basic theory of Faraday waves respectively,and provide a series of important equations describing fluid dynamics and Mathieu equation describing parametric resonance.The dispersion effect of fluid is introduced,which demonstrates that it is difficult for traveling waves to remain stable for a long time in fluid.The properties of the solution of Mathew equation directly lead to the occurrence of parametric resonance.This is the basis of understanding and studying Faraday waves,the essence of which is a parametric resonance phenomenon.By understanding the properties of fluid mechanics and Mathew equation,we can better explain the generation mechanism of Faraday waves.At the same time,the number of eigen-modes excited at the corresponding driving frequency is judged.In Chapter 3,experimental equipment and waveform extraction methods are introduced.The waveform information can be obtained accurately through simple design.At the same time,the phenomenological description of the experimental phenomenon is also carried out.By observing the deformation of the wave peak,it is obvious that there are soliton-like waves moving back and forth in the flume tank.Finally,using the waveform data obtained after processing,the evolution diagram of the whole waveform with time is given.In Chapter 4,the waveform is analyzed by discrete cosine transform and Fourier transform.The basis of discrete cosine transform is the eigen mode of waveform in the long water tank.Through this transformation,the components of each eigen mode in waveform can be clearly obtained.With the results of theoretical calculation,two modes(14,0)and(11,1)are dominant.DCT directly corresponds to the intrinsic mode of the system and does not contain phase information.So we use Fourier transform to get the phase information of each mode under this transformation.In Chapter 5,we use Singular Value Decomposition(SVD)algorithm to process the massive and miscellaneous waveform data,and successfully separate the standing wave components from traveling wave components in this phenomenon.The main modes of the standing wave components and the corresponding modes of the traveling wave are determined by Fourier analysis of the SVD decomposed parts.At the same time,it also benefits from the difference between the two in the time cycle.Using SVD algorithm,we can get the main information of the whole system only from the data level,without relying on each person’s intuition and experience,which is the biggest advantage of SVD algorithm.This paper introduces Sparse Identification of Nonlinear Dynamics(SINDy)algorithm in machine learning and gives a solution to this problem.Finally,I summarize this thesis and make a prospect for the future research work.