Scattering of gravity waves by a periodically structured ridge of finite extent

Abstract

We study the propagation of water waves over a ridge structured at the subwavelength scale using homogenization techniques able to account for its finite extent. The calculations are conducted in the time domain considering the full three-dimensional problem to capture the effects of the evanescent field in the water channel over the structured ridge and at its boundaries. This provides an effective two-dimensional wave equation which is a classical result but also non-intuitive transmission conditions between the region of the ridge and the surrounding regions of constant immersion depth. Numerical results provide evidence that the scattering properties of a structured ridge can be strongly influenced by the evanescent fields, a fact which is accurately captured by the homogenized model.

Time domain modelling of a Helmholtz resonator analogue for water waves

“…Next, the resonant amplitude |φ R (ω 0 )| increases with H for small H in agreement with (3.5), afterwards, it saturates (above H 40 m in the main plot in (a)). In the shallow water regime, the increase of the resonant amplitude with H tells us that the presence of the sea bottom below the cavity weakens the resonance; a similar confinement effect has been reported for acoustic HRs (Maurel et al 2019). However, as the water depth increases, this effect vanishes and the amplitude tends to a value corresponding to the resonance in the deep water regime (hence, independent of H).…”

“…For instance, the propagation can be made anisotropic using varying bathymetry, see e.g. Maurel et al (2017), Maurel, Pham & Marigo (2019) and Porter (2019), or it can be guided due to valley-locked transport (Makwana et al 2020). In addition, a plethora of interesting phenomena has been proposed, such as the cancellation of the scattering by rigid obstacles and their cloaking (Newman 2014;Dupont et al 2016;Porter 2018;Bobinski et al 2018;Iida & Kashiwagi 2018;Farhat et al 2020), the perfect absorption of the wave energy in the nonlinear regime (Monsalve et al 2019) and the trapping for energy harvesting using graded arrays of resonators (Bennetts, Peter & Craster 2018, 2019.…”

Control of the Swell by an Array of Helmholtz Resonators

“…More generally, the modern design of devices able to control the energy flow of ocean waves has begun to benefit from the development of metamaterials. For instance, the propagation can be made anisotropic using varying bathymetry see e.g., [3][4][5], or it can be guided due to valley-locked transport [6]. In addition, a plethora of interesting phenomena has been proposed, such as the cancellation of the scattering by rigid obstacles and their cloaking [7][8][9][10][11][12], the perfect absorption of the wave energy in the nonlinear regime [13] and the trapping for energy harvesting using graded arrays of resonators [14,15].…”

Shallow water theory for structured bathymetry

“…More recently, Maurel et al [15] have considered shallow water wave scattering by a structured bed of finite width by adopting the model of Maurel et al [14], with a particular focus on determining effective matching conditions at the discontinuity between a conventional bed and a structured bed. This is done through a careful matched asymptotic process.…”