Bragg scattering of long waves by an array of floating flexible plates in the presence of multiple submerged trenches

Abstract

Bragg scattering of long gravity waves by an array of floating elastic plates in the presence of a series of rectangular trenches is studied under the assumption of linearized water wave theory and small amplitude structural response. Bragg reflection occurs in the case of an array of multiple floating flexible plates and trenches in isolation or combination, and the number of sub-harmonic peaks between two consecutive peaks is two less than the number of plates/trenches. In the case of long-wave scattering by an array of trenches in the absence of floating plates, wave reflection becomes zero minimum for a certain fixed wavenumber at the end of the first cycle (as referred to 0 < k1h1 < 0.32), irrespective of the even/odd number of trenches. Furthermore, the Bragg resonant reflection pattern of the second cycle is a mirror image of the reflection occurring in the first cycle. Conversely, the symmetric pattern of Bragg reflection exhibited in the case of an array of trenches does not occur for an array of plates in isolation or for plates and trenches in combination. Between consecutive harmonic peaks, the minima/maxima in the wave reflection occur in the case of the even/odd number of trenches/plates in isolation. In contrast, it occurs within the first cycle in the case of trenches and plates in combination. Between consecutive harmonic peaks, maxima/minima of wave reflection coincide for a certain wavenumber, irrespective of the even/odd number of trenches/plates in isolation. In addition, these maxima attained for a certain wavenumber for the even number of trenches/plates are 180° out of phase to that of minima for an odd number of trenches/plates. However, similar phenomena occur within the first cycle of Bragg resonance in the case of trenches and plates in combination. Moreover, the amplitude of oscillation in the wave reflection increases with an increase in plate rigidity. Time-dependent motion due to Bragg scattering by trenches and plates is demonstrated in different cases.

Bragg Scattering of Surface Gravity Waves Due to Multiple Bottom Undulations and a Semi-Infinite Floating Flexible Structure

“…In this section, using MATLAB software, numerical codes are written to analyze the effect of a semi-infinite floating plate on Bragg scattering of surface waves by multiple undulations in the seabed. Various non-dimensional physical parameters, such as width w/L 1 , gap between the leading edges of two adjacent trenches/breakwaters 2d/L 1 of trench/breakwater, gap between right-most trench/breakwater, and plate L g /L 1 , are considered for the analysis of Bragg scattering of gravity waves, as in Kar et al [18]. Moreover, the values of different physical parameters, such as water depth h 1 = 5 m, incident wavelength L 1 (= 2π/k 1 ) = 40π m, and plate rigidity D/ρg = 10 5 m 4 (same as considered by Kalyanaraman et al [12]), and mass density of the plate is m s = ρ i d i , with density of ice ρ i = 922 kg m −3 , plate thickness d i = 1 m, density of water ρ = 1025 kg m −3 , and g = 9.81 ms −2 are kept fixed, unless stated otherwise.…”

“…where K = ω 2 /g is the wavenumber associated with the surface gravity waves in infinite water depth. Further, on the plate covered surface, the linearized dynamic boundary condition yields (see Stoker [25] and Kar et al [18] for details)…”

“…The long wave equation in the open water region in term of φ 1 (x) is given by (see Kar et al [18] and Sahoo [28] for details)…”

“….N. The linearized long wave equation in the plate covered region in term of the spatial component φ 2 (x) is given by (see Kar et al [18])…”

“…The study of Kar et al [17] reveals that the maximal peak Bragg reflection curve is nearly linear in the case of long wave scattering by a pair of submerged trenches, whilst the pattern of the said curves appears to be non-linear in the case of an array of finite number of submerged trenches. Kar et al [18] studied the scattering of long gravity waves due to an array of submerged trenches and breakwaters in the presence of an array of floating flexible plates.…”