Bragg Resonance And Bloch Band Gap Of Linear Water Waves Caused By Rectified Cosinoidal And Parabolic Trenches

Bragg resonance in water waves has attracted a lot of attention since it was first discovered by British scholar Davies in 1982.The earliest studies mainly focused on the Bragg resonance caused by sinusoidal ripples and the influence of standing waves caused by Bragg resonance on the further development of sinusoidal ripple itself.Then,after Mei,a member of the American Academy of Engineering,introduced the concept of artificial bars in 1988,scholars began to study Bragg resonance caused by artificial bars and their possible role in bank protection.In the last couple of years,researchers have begun to study Bragg resonance caused by a finite periodic array of trenches.In the first part of this dissertation,we study the wave Bragg resonance excited by a series of periodic trenches consisting of a finite number of parabolic or rectified-cosine trenches respectively.By means of variable substitution,the modified mild-slope equation(MMSE)with the coefficients being implicit functions is firstly transformed into an explicit equation with the coefficients being explicit function.Then,the Frobenius series solution of the modified mild-slope equation is constructed,and the convergence condition of the series solution is given.Finally,an analytical formula of reflection coefficient is established by using the coupling condition of mass conservation.According to the analytical formula of reflection coefficient,the influence of the number of trenches,the dimensionless width and height(relative to the incident wavelength)on the resonance peak,the resonance phase and resonance bandwidth is analyzed.The results show that when the width and height of trenches are fixed,the resonance peak value gradually increases and tends to 1 with the increase of the number of trenches,while the resonance bandwidth gradually narrows and tends to a fixed value.When the other parameters are fixed,Bragg resonance peak increases gradually with the increase of the trench depth.When the other parameters are fixed,the reflection coefficient is a periodic function of the dimensionless width of trenches.Especially,with the increase of the trench width,Bragg resonance peak increases first and then decreases,which indicates that there is a particular trench width to make the resonance peak reach the maximum,which lays a theoretical foundation for the optimization of the Bragg resonance reflection with respect to the trench width.In particular,the phase upshift of resonance peak,which was recently observed in a periodic array of cycloidal trenches,has been confirmed again in a periodic array of parabolic trenches and rectified cosinoidal trenches studied in this thesis,indicating that the phase upshift of resonance peak is a common phenomenon for wave reflection by periodic trenches.As the second major part of this thesis,we study the bandwidth of Bragg resonance by means of the band theory of water waves on the infinite period seabed topography.Firstly,by means of the analytical solution of the MMSE constructed in the first part,and combined with Bloch's theorem,we deriveed the band expression(also known as the dispersion relation)of Bloch waves over the infinite periodic array of parabolic trenches and rectified cosinoidal trenches,respectively.The calculated results show that Bragg resonance band is almost one-to-one corresponding to the Bloch bandgap,which indicates that the analytical formula of reflection coefficient and the analytical formula of Bloch band expression confirm each other.Based on the Bloch band expression,the influence of the depth and width of these two types of trenches on the width of Bloch bandgaps is analyzed and discussed.