Bragg Reflection Of Linear Water Waves By Cycloidal Bragg Breakwaters And Trenches

The so-called Bragg breakwaters consist of several submerged artificial bars parallel to the coastline with an identical shape,size and distance.According to the Bragg’s law,once the distance between two adjacent bars is about half of the wavelength of incident waves,Bragg resonant reflection occurs,which leads to a wall of water to prevent the invasion of the incident wave train to the coast.Therefore,it is significant to investigate wave reflection by Bragg breakwaters.In this master’s thesis,based on the modified mild-slope equation(MMSE),the propagation of linear water waves over a finite array of cycloidal bars or trenches is investigated.By using variable transformation,series-solution technique and matrix multiplication,wave reflection is modeled analytically.On one hand,since the the MMSE is to too complicated,it is impossible to give an analytical solution in a closed form.On the other hand,since the physical domain occupied by bars or trenches contains a singular point to the MMSE,it is also impossible to construct an analytical solution in terms of Taylor series.We turn to seek a series solution to the MMSE in terms of Frobenius series and it worked.The convergence condition of series solution is clarified by using both algebraic inequalities and geometrical graphs.Using the present series solution,the influence of the number,width and height of cycloidal bars and trenches on Bragg resonance is analyzed.It is shown that,for fixed width and height of bars or trenches,reflection coefficient is a periodic function of dimensionless distance.When the total number of bars or trenches increases,the resonance peak value increases but the resonance bandwidth decreases.When the width of bars or trenches increases,the peak value of resonance reflection first increases and then decreases,which means that the peak value can be maximized at a fixed bar width.In addition,as the bar height increases,its peak value also increases gradually.It has been shown by many previous studies that Bragg resonance does not obey Bragg’s law,specifically speaking,the resonance does not occur exactly at the phase with the bar distance being half of the incident wavelength,while occurs at a phase with the bar distance being less than the half of incident wavelength,which was named as phase downward shift.In the preset study,for cycloidal bars,this phase downward shift still exist.However,it is very interesting in the present studies,for cycloidal trenches,Bragg resonance occurs at a phase with the bar distance being larger than the half of incident wavelength,that is,the phase shifts upward to a high frequency.Finally,for wave reflection by a single bar,by comparing the present analytical solution to the MMSE with two approximate solutions to the MMSE given by several Mexican scholars,it is found that both of their results are wrong.Actually,a solution in terms of Taylor series cannot exist due to a singularity of the MMSE in the physical domain occupied by the cycloidal bar even though it was formally given,which leads to a divergency of their two solutions.