Rainbow reflection of linear long waves excited by a finite graded array of trapezoidal bars

In this paper, the Bragg reflection and rainbow reflection/trapping of linear long (or shallow-water) waves excited by a finite array of trapezoidal bars are studied from the perspective of Bloch band theory. Firstly, a closed-form solution of the reflection coefficient for wave propagation over a finite non-uniform array of trapezoidal bars is derived. Secondly, the relation between Bloch band gaps modulated by an infinite uniform array of trapezoidal bars and the Bragg reflection excited by the cognate finite array is investigated. It is revealed for the first time that there is a strong positive correlation between the width of the $n$th order Bloch band gap and the intensity of the $n$th order Bragg resonance. Thirdly, the rainbow reflection of linear long waves excited by a finite graded array of trapezoidal bars is studied, and the arrangement of bar spacing for rainbow reflection over a broad and continuous bandwidth is proposed. In addition, the strength of rainbow reflection can be enhanced by increasing the number of bars and the filling fraction of bars. Finally, for a target frequency range that needs to be blocked in practical applications, the Bloch band structures could be applied to guide the layout of bars for rainbow reflection.