Trapping of water waves by moored bodies

Abstract

Certain types of floating bodies are known to support trapped modes, with oscillatory fluid motion near the body and no energy radiation in the far field. Previous work has considered either fixed bodies, where the boundary conditions are homogeneous, or bodies which are freely floating and moving without any exciting force. For a fixed body the existence of a trapped mode implies that there is no unique solution of the boundary-value problem for the velocity potential with a prescribed body motion. For a free body which supports a trapped mode, the solution of the coupled problem for the motions of the fluid and body does not have a unique solution. A more general case is considered here, of a body with a linear restoring force such as an elastic mooring. The limiting cases of a fixed and free body correspond to infinite or zero values of the corresponding spring constant. A variety of body shapes are found including cylinders in two dimensions and axisymmetric bodies in three dimensions, which illustrate this more general case of trapping and provide a connection between the fixed and free cases.

Radiation, trapping and near-trapping in arrays of floating truncated cylinders

“…Porter and Evans [26] started with a pair of rectangular cylinders or a thick-walled cylindrical shell (in 2-and 3-dimensions respectively) and varied the geometry to find motion trapping structures. Motion-trapping structures with mooring restraints were studied by Evans and Porter [5], who found that a moored submerged circular cylinder moving in heave or sway could be a motion trapping structure, and Newman [25] who analysed mooring stiffnesses on the continuum from −∞ to ∞.…”

Freely floating structures trapping time-harmonic water waves

“…Indeed, ϕ m satisfies the same estimates at infinity as the remainder R in (13). Thus, any function ϕ m defined by formulae (18) and (19) with r = r m can serve as the first component of an eigensolution provided the vectors χ (k) * and the water domain W are chosen properly.…”

“…No other rigorous results about this problem had been obtained until recently. However, after 2005 a number of authors considered the question of trapped modes at the heuristic level and various two-dimensional and axisymmetric trapping structures were proposed by virtue of numerical computations (see [13,14,3,19,20,21,4], which are listed in the chronological order). In most of these papers, a simplified model is treated; it deals with a freely floating body constrained to the heave motion only.…”

Passive trapped modes in the water-wave problem for a floating structure

“…The right-hand side of equation 12is proportional to the hydrodynamic force on the structure due to the fluid oscillations. For a moored structure, the possibility of finite-energy motions that satisfy (11) and (12) with non-zero φ and V has been known for some time and there have been a number of recent developments (Evans & Porter 2007, Newman 2008. Non-trivial solutions with finite energy of equations (9)-(12) for structures without moorings are referred to as motion trapped modes and, for V = 0, their existence has been established using the inverse method by McIver & McIver (2006, and solutions for half-immersed circular cylinders have been found by Porter & Evans (2009).…”