A new law of Class I Bragg resonance for linear long waves over arrays of parabolic trenches or rectified cosinoidal trenches

Class I Bragg resonances of linear long waves excited by a finite periodic array of widely spaced trenches were studied, where the trenches include two types, i.e., parabolic trench and rectified cosinoidal trench. First, analytical solutions of the reflection coefficient in terms of the associated Legendre functions for wave reflection by parabolic trenches and in terms of Frobenius series for wave reflection by rectified cosinoidal trenches are constructed, respectively. Second, the phenomenon of the phase upshift of each order Bragg resonance is confirmed for linear long waves reflected by the two types of trenches. Third, the dispersion relation of linear long waves over an infinite periodic array of parabolic or rectified cosinoidal trenches is derived. Fourth, based on the dispersion relation, Bloch band structures are obtained and their related periodic gap maps are plotted. Finally, based on the Bloch band theory, a new law for locating the phase of each order Bragg resonance is established, which is much more accurate than the traditional Bragg’s law. By using the new law, the phase upshift phenomenon can be well explained, visually demonstrated and accurately predicted.