Cloaking of a vertical cylinder in waves using variable bathymetry

Abstract

The paper describes a process which allows a vertical circular cylinder subject to plane monochromatic surface gravity waves to appear invisible to the far-field observer. This is achieved by surrounding the cylinder with an annular region of variable bathymetry. Two approaches are taken to investigate this effect. First a mild-slope approximation is applied to the governing linearised three-dimensional water wave equations to formulate a depth-averaged two-dimensional wave equation with varying wavenumber over the variable bathmetry. This is then solved by formulating a domain integral equation, solved numerically by discretisation. For a given set of geometrical and wave parameters, the bathymetry is selected by a numerical optimisation process and it is shown that the scattering cross-section is reduced towards zero with increasing refinement of the bathymetry. A fully three-dimensional boundary element method, based on the WAMIT solver but adapted here to allow for depressions in the bed, is used to assess the accuracy of the mild-slope results and then further numerically optimise the bathymetry towards a cloaking structure. Numerical results provide strong evidence that perfect cloaking is possible for the fully three-dimensional problem. One practical application of the results is that cloaking implies a reduced mean drift force on the cylinder.

Revisiting the anisotropy of metamaterials for water waves

“…The design of metamaterials for water waves has generated in the past 10 years an increasing interest for applications including the realization of lenses [1][2][3], the control of the ocean wave energy flow [4][5][6] or the cloaking able to produce a protected free wave region [7][8][9][10][11][12][13]. For most of the cases, these metamaterials require anisotropic effective parameters that can take extreme values, a task that can be challenging.…”

Cloaking a vertical cylinder via homogenization in the mild-slope equation

“…Since then, some studies have been carried out in order to control water wave propagation. Alam (2012) considered stratified seas to cloak floating objects; Berraquero et al (2013) realized a water wave shifter; Newman (2014) used additional bodies with optimized geometries surrounding a cylinder to give numerical evidence for perfect cloaking; McIver (2014) proved that perfect cloaking cannot be achieved with fixed bodies, floating bodies or variations in the seabed topography; Porter & Newman (2014) proposed a numerical optimization process over bathymetry around a circular cylinder in order to achieve near-perfect cloaking for linear waves; Dupont et al (2015) studied an invisibility carpet in a wavetank for linear waves; recently, Zareei & Alam (2015) used nonlinear transformation to achieve cloaking in shallow-water waves.…”

“…The final cloaking device is constructed using a conformal map. We perform numerical investigation using the finite element method (FEM) to show that we can approach a cloaking effect with non-ideal parameters and a reduction of the mean drift force (see Porter & Newman 2014). …”

The scattering properties of a system of structures in water waves

“…The concept of cloaking has also been explored in acoustics (4), surface waves (5) and elasticity (6), (7) and a special issue of Wave Motion is devoted to work which discusses cloaking in a variety of physical situations (8). In linear water-wave theory Porter and Newman (9), (10), (11), (12) use the term cloaking to describe the elimination of the far-field scattered waves that arise when a monochromatic plane wave is incident upon a fixed, rigid structure and this is the definition that will be used in this work. In particular, they investigated whether is is possible to cloak a vertical circular cylinder in water waves, by either varying the sea-bed topography in the neighbourhood of the cylinder or by surrounding it with other structures.…”