A Study of Transverse Waves

This dissertation, "A Study of Transverse Waves" by Limei, Deng, 鄧麗梅, was obtained from The University of Hong Kong (Pokfulam, Hong Kong) and is being sold pursuant to Creative Commons: Attribution 3.0 Hong Kong License. The content of this dissertation has not been altered in any way. We have altered the formatting in order to facilitate the ease of printing and reading of the dissertation. All rights not granted by the above license are retained by the author. Abstract: Abstract of thesis entitled A Study of Transverse Waves submitted by Li-Mei Deng for the degree of Doctor of Philosophy at the University of Hong Kong in December 2005 Transverse waves are waves that have crest lines perpendicular to the wavemaker and have the same frequency as that of incident waves. Transverse waves have been shown to appear in the cross section of a water channel when the incident-wave wavelength is less than or equal to twice the channel width. In this thesis, transverse waves are further studied experimentally and theoretically in order to investigate their occurrence conditions and occurrence mechanism. A series of experiments are conducted separately in a channel with rectangular cross sections and constant width, in a channel with decreasing width, and in a channel with an addition of a false bottom. The characteristics and the occurrence conditions of transverse waves are obtained in a rectangular channel. During the experiments, first three modes of transverse waves were observed despite the limitation of channel width, and the mode frequencies are found to be close to the frequencies that satisfy the dispersion relation. With the variation of channel width and water depth and the introduction of reflection structures, the occurrence conditions of transverse waves are different. Transverse waves can be relatively easily captured in a channel without absorbing structures and in a channel with a false bottom. The maximum amplitudes of stable transverse waves are estimated by the Transfer Function Method (TFM). The relationship between the driving amplitude and the occurrence of transverse waves and the growth rate of transverse waves are also obtained. Based on the previous observations, there exists a time delay between the occurrence of transverse waves and the propagation of incident waves, and transverse waves generally appear near the wavemaker first. Therefore, transverse waves are assumed to occur due to a small perturbation near the wavemaker. Based on the perturbation method, theoretical analysis is performed in order to study the mechanism of transverse waves in a finitely long channel with a finite water depth. The evolution equation deduced from the Laplace equation and boundary conditions is shown to be a nonlinear equation with boundary conditions. Different order equations are analyzed for the occurrence of transverse waves. The nonlinear terms retained in the governing equation are identified to play a key role in the energy transfer and the existence of transverse waves. The formulations of the first-order and second-order problems are analyzed and compared with experimental results. Finally, the resonance mechanism of transverse waves is discussed and the occurrence conditions of transverse waves in a finitely long channel are proposed. Transverse waves are considered as an example of parametric resonance because their appearance is dependent on the driving amplitude. The existence of critical driving amplitude above which transverse waves occur coincides with the experimental observations. The nonlinear relationship between the driving amplitude and the occurrence frequency and the appearance of transverse waves near the mode frequencies also validate the mechanism of transverse waves. DOI: 10.5353/th_b3627190 Subjects: Shear waves