Analytical study of Bragg resonances by a finite periodic array of congruent trapezoidal bars or trenches on a sloping seabed

Abstract

For water waves propagating over a finite periodic array of congruent trapezoidal bars on a sloping seabed or trenches dug on a sloping seabed, an analytical solution in terms of Taylor series to the modified mild-slope equation (MMSE) is constructed. Based on the analytical solution, influences of various topographic parameters (e.g., the slope of seabed, bar/trench number, bar height or trench depth, and bar/trench width) on the peak value, peak phase and zero reflection of Bragg resonances are investigated. Especially, starting from the original definition of Bragg resonances, a modified Bragg's law for characterizing the excitation condition of the nth order Bragg resonance is derived, which shows that the excitation of Bragg resonances depends entirely on the average phase velocity of the water wave within the bar/trench field. When the influence of the bar/trench field on the average phase velocity is quite small, the modified Bragg's law degenerates into the traditional Bragg’s law borrowed from X-ray crystallography, which also reflects the essential difference between the water wave as a mechanical wave and the non-mechanical X-ray wave. In addition, for N bars/trenches on a flat seabed, there are at least N−1 zero reflections between any two adjacent Bragg resonance peaks. When the bars are very low or the trenches are very shallow, the phase of the N−1 zero reflections in those zero reflections is exactly the position of the N−1 zero points of the Chebyshev polynomial of the second kind, UN−1(cosk0d), where k0 is the wavenumber on the flat seabed and d is the bar/trench spacing. However, once a sloping seabed is added, all these N−1 zero reflections disappear possibly due to the destruction of the symmetry of the background bar/trench field. Finally, regardless of the number of bars/trenches, when the bar/trench width varies, the peak value of the nth order Bragg resonance oscillates periodically, leading to n local maxima. A comparative study shows that these characteristics are consistent with the band structure shown in the Bloch periodic gap map (PGM).