A modified Bragg's law of Class I Bragg resonances of linear long waves excited by an array of artificial bars

TL;DR

This paper develops a modified Bragg's law for linear long water waves reflecting off finite periodic artificial bars, accounting for bar shape and size, and explains phase downshifting phenomena.

Contribution

It introduces a new modified Bragg's law that incorporates bar shape and size, extending classical Bragg's law to finite artificial bar arrays in water wave contexts.

Findings

Modified Bragg's law accounts for bar shape and size.

Bragg's law is a limiting case when bar dimensions approach zero.

Phase downshifting increases with bar cross-sectional area.

Abstract

Based on the band theory of Bloch waves, a modified Bragg's law is established for Class I Bragg resonances when linear long waves are reflected by a finite periodic array of artificial bars, including rectangular, parabolic, rectified cosinoidal, isosceles trapezoidal, and isosceles triangular bars. The modified Bragg's 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 is just a limiting case of the modified Bragg's law as the bar height or width approaches zero. This fact tells us that the Bragg's law that accurately describes Bragg resonances in X-ray crystallography cannot be simply applied to Bragg resonances of linear long water waves excited by any finite periodic array of artificial bars, because the height and width of any artificial bar cannot be zero. Based on the modified Bragg's 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.