The Band Structure Of Shallow Water Waves Propagating Over Infinite Series Of Periodic Truncated-cone

Since 1990's, the energy band theory, which had widely used in solid mechanics,has been extended to study the propagation of liquid surface waves over infinite array of periodic bottom structures and much achievement has been obtained. However,during these more than twenty years, almost all the related researches are restricted in simple bottom structures with piecewise constant depth such as two-dimensional rectangular steps and three-dimensional(water piercing, submerged or bottom drilled)vertical circular cylinders or polygonal cylinders. The infinite array of periodic bottom structures was rarely studied, since a differential equation with variable coefficients needs to be solved if the linear shallow-water equation is adopted as governing equation,or even a differential equation with implicit variable coefficients needs to be solved if the Ye's equation or the mild-slope equation is adopted as governing equation due the implicit linear dispersion relation.In this thesis, the band structure of linear shallow-water waves over infinite arrays of periodic truncated-cones is studied. It is clear that the water depth over the truncatedcones varies continuously, called variable water depth, which has been rarely discussed in previous research. The linear shallow-water equation(or the linear long-wave equation)is chosen as the governing equation. Based on the theory of Fourier series expansion,the periodic water depth function is expanded into Fourier series, and the periodic part of the free surface elevation is also expanded into Fourier series with the expansion coefficients to be determined. By substituting these two Fourier series into shallowwater equation and by truncating the infinite summation into a finite summation, the original problem of band structures is transformed into a matrix eigenvalue problem.Finally, the band structures of shallow water waves over an infinite array of periodic truncated-cones arranged in the pattern of square lattice and hexagonal lattice are given,and some complete band gaps are found which mean that the propagation of waves with the frequency within the gaps are absolutely forbidden. Further, the influence of various topography parameters such as the fillings of the top and bottom planes of the truncated cones on the width and location of the band gaps is analyzed. The present results have theoretical value for the practical construction of finite arrays of periodic structures.