An approximate law of Class I Bragg resonance of linear shallow-water waves excited by five types of artificial bars

Abstract

Based on the band theory of Bloch waves, an approximate law is established for locating the phase of Class I Bragg resonance when linear shallow-water waves are reflected by a finite periodic array of widely spaced artificial bars, including rectangular, parabolic, rectified cosinoidal, isosceles trapezoidal, and isosceles triangular bars. The approximate law is described by a function of several parameters such as the bar shape, dimensionless bar height with respect to the global water depth, and dimensionless bar width with respect to the incident wavelength. It is found that Bragg’s law that accurately describes Bragg resonances in X-ray crystallography cannot be simply applied to Bragg resonances of linear shallow-water waves excited by any finite periodic array of widely spaced bars, because the height and width of any bar is nonzero. Based on the approximate law, the phenomenon of phase downshifting can be well explained and accurately predicted. It is revealed that the phase downshifting becomes more significant with increasing the cross-sectional area of artificial bars.