On the problem of time-harmonic water waves in the presence of a freely floating structure

Abstract

Abstract. The two-dimensional problem of time-harmonic water waves in the presence of a freely floating structure (it consists of a finite number of infinitely long surface-piercing cylinders connected above the water surface) is considered. The coupled spectral boundary value problem modeling the small-amplitude motion of this mechanical system involves the spectral parameter, the frequency of oscillations, which appears in the boundary conditions as well as in the equations governing the structure's motion. It is proved that any value of the frequency turns out to be an eigenvalue of the problem for a particular structure obtained with the help of the so-called inverse procedure. §1. IntroductionThis paper deals with coupled boundary value problems describing the irrotational motion of an inviscid, incompressible, heavy fluid (water) in which a structure consisting of a finite number of rigid bodies is floating freely. Water extends to infinity in the horizontal directions as well as downwards and is bounded above by a free surface, thus modeling an infinitely deep open sea. The surface tension is neglected on the free surface and the coupled motion of the structure and water is assumed to be of small amplitude near equilibrium, which allows us to use a linear model. We restrict ourselves to considering structures formed by infinitely long horizontal surface-piercing cylinders that are rigidly connected with each other above the water surface (an infinitely long pontoon is a typical example). This assumption leads to another simplification that consists in studying only the case of two-dimensional motion that is the same in every plane orthogonal to the structure's generators.The time-dependent equations for the two-dimensional mechanical system described above are formulated in Subsection 2.1. A full discussion of the three-dimensional coupled boundary value problem was given by John in his pioneering work [1] (see also the paper [2] by Beale, who presented the equations of motion in a more convenient matrix form). In Subsection 2.2, we turn to the case where the water waves are time-harmonic, as well as the motion of the structure, and the so-called external forces (they, for example, are due to constraints on the structure motion) are absent. We also discuss conditions that must be imposed on the behavior of a time-harmonic solution at infinity. The resulting problem is the coupled spectral problem involving the spectral parameter, the frequency of oscillations, which appears in the boundary conditions as well as in the equations of 2010 Mathematics Subject Classification. Primary 76B15, 76B03, Secondary 35Q35, 35P05.

On the coupled time-harmonic motion of a freely floating body and water covered by brash ice

“…In order to obtain a family of immersed parts of motionless bodies that trap waves in the water covered by the brash ice we follow the considerations used in § § 3 and 4 of Kuznetsov (2011) with ν changed to ν (σ ) = ν/(1 − νσ ).…”

“…It should be mentioned that the same semi-inverse procedure was used by Kuznetsov (2011) for obtaining two-dimensional trapping bodies that are motionless in the open water. However, there is an essential distinction between the case of open water and water covered by brash ice, namely no restriction was imposed on the trapping frequencies in the former case.…”

The singularity expansion method and near-trapping of linear water waves

“…The existence of such modes was later demonstrated for freely floating bodies constrained to move in heave (McIver & McIver 2006;Fitzgerald & McIver 2010). More recently, the existence of these modes in the floating-body problem was rigorously proved by Kuznetsov (2011) and Kuznetsov & Motygin (2012 without restrictions on the mode of body motion. Trapped modes are mainly of theoretical importance since they prove non-uniqueness in the water wave problem.…”

Trapping of water waves by freely floating structures in a channel

“…In 1950, John proved that the solution to the radiation problem is unique if the frequency of the harmonic motion is large enough and the freely floating object satisfies a certain geometrical condition (no point of the immersed surface lies below a point of the free surface). Since then, the problem of trapping of water waves by freely floating structures has rarely been considered in full generality, notable exceptions being the articles of Beale (1977), who studied the corresponding initial-value problem, and Kuznetsov (2010), who considered the two-dimensional problem and observed that motionless freely floating structures support trapped mode solutions. In most recent work, attention has been either restricted to the two-dimensional case (McIver & McIver 2006;Evans & Porter 2007;Porter & Evans 2009) or the motion of the floating structure constrained to heaving motion Newman 2008;Porter & Evans 2008).…”

“…Indeed, a reduction scheme in the form of a suitable trace operator (cf. Nazarov 2008) has led to a series of simple sufficient conditions guaranteeing the existence of trapped modes around fixed structures in different geometrical configurations (see Nazarov 2009a-c;Nazarov & Videman 2009, 2010.…”