Linear Water Waves: A Mathematical Approach

Kuznetsov, Nikolay; Maz′ya, Vladimir; Vainberg, Boris; Miles, J.P.

doi:10.1115/1.1553438

Cited by 83 publications

(190 citation statements)

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“…The progress achieved in this direction was presented in detail in the book [6,Part 1]. In particular, much attention was given to the existence of trapped modes in the case of a fixed surfacepiercing structure (the groundbreaking paper [7] on this topic was published by M. McIver in 1996).…”

Section: §1 Introductionmentioning

confidence: 99%

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

Kuznetsov

2011

St. Petersburg Math. J.

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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.

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Section: §1 Introductionmentioning

confidence: 99%

“…We recall that, in accordance with the standard linearization procedure (see [1, §2] and [6,Introduction]), all unknowns are proportional to a small nondimensional parameter (in fact, expansions of unknowns in powers of are used). In the linearized problem, the boundary conditions are imposed on ∂W .…”

Section: §1 Introductionmentioning

confidence: 99%

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

Kuznetsov

2011

St. Petersburg Math. J.

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“…As the system of structures satisfies the John condition, it is convenient to follow (16) and define 13) at all points (x, y) on F . The two-dimensional Laplacian of w is taken and after some manipulation with the use of the governing equation and boundary conditions for φ, it may be shown that w satisfies the two-dimensional Helmholtz equation (3.5).…”

Section: A System Of Three-dimensional Structuresmentioning

confidence: 99%

The scattering properties of a system of structures in water waves

McIver

2014

The Quarterly Journal of Mechanics and Applied Mathematics

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“…[8], we introduce the single-valued velocity potentials Φ (1) (x, y, z, t) and Φ (2) (x, y, z, t) in the upper and lower fluid regions of the channel and the vector a ∈ R 6 describing, at each instant of time t, the displacement of the centre of mass of the body from its rest position x 0 (components a j , j = 1, 2, 4, respectively, for surge, sway and heave) and the angular displacement of the body about the axis through the centre of mass x 0 (components a j , j = 3, 5, 6, respectively, for yaw, roll and pitch).…”

Section: Equations Of Motionmentioning

confidence: 99%

“…Yet, the linearized equations describing the small amplitude motion of the coupled system consisting of an inviscid, incompressible, heavy fluid and a rigid body floating freely in it were derived only in the late 1940s by John [3]. The classical waterwave problem where an immersed obstacle is assumed to be fixed was thoroughly investigated over the past century (see [4][5][6][7][8][9] and references therein) but the coupled freely floating problem was largely ignored until recently.…”

Section: Introductionmentioning

confidence: 99%

Trapped modes around freely floating bodies in a two-layer fluid channel

2014

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Unlike the trapping of time-harmonic water waves by fixed obstacles, the oscillation of freely floating structures gives rise to a complex nonlinear spectral problem. Still, through a convenient elimination scheme the system simplifies to a linear spectral problem for a self-adjoint operator in a Hilbert space. Under symmetry assumptions on the geometry of the fluid domain, we present conditions guaranteeing the existence of trapped modes in a two-layer fluid channel. Numerous examples of floating bodies supporting trapped modes are given.

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Linear Water Waves: A Mathematical Approach

Cited by 83 publications

References 1 publication

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

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

The scattering properties of a system of structures in water waves

Trapped modes around freely floating bodies in a two-layer fluid channel

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