Bloch band gap of shallow-water waves over infinite arrays of parabolic bars and rectified cosinoidal bars and Bragg resonance over finite arrays of bars

Abstract

In this paper, eigenvalue problems of Bloch waves over an infinite array of equally spaced parabolic bars and rectified cosinoidal bars in shallow water are respectively studied. Both band expressions in closed form for the two eigenvalue problems are derived, which depend upon the dimensionless bar height with respect to water depth and the dimensionless bar width and spacing with respect to incident wavelength. It is found that the band-gap center associated with an infinite array of bars coincides well with the peak phase of Bragg resonance by any finite array of bars. For any fixed dimensionless bar height, a gap map as a bivariate function of the dimensionless bar width and spacing can be given. When both the dimensionless bar height and width are fixed, the band gap occurs periodically. When the dimensionless bar height is fixed, there exists a particular dimensionless bar width such that the band gap achieves its maximal width. Based on the gap maps, the well-known phenomenon of phase downshift can be visually and quantitatively explained, and the peak phase of Bragg resonant reflection by a finite array of bars can be well predicted, which is much more accurate than that predicted by Bragg’s law.