Freely floating structures trapping time-harmonic water waves

Abstract

We study the coupled small-amplitude motion of the mechanical system consisting of infinitely deep water and a structure immersed in it. The former is bounded above by a free surface, whereas the latter is formed by an arbitrary finite number of surface-piercing bodies floating freely. The mathematical model of time-harmonic motion is a spectral problem in which the frequency of oscillations serves as the spectral parameter. It is proved that there exist axisymmetric structures consisting of N ≥ 2 bodies; every structure has the following properties: (i) a time-harmonic wave mode is trapped by it; (ii) some of its bodies (may be none) are motionless, whereas the rest of the bodies (may be none) are heaving at the same frequency as water. The construction of these structures is based on a generalization of the semi-inverse procedure applied earlier for obtaining trapping bodies that are motionless although float freely.

On freely floating bodies trapping time-harmonic waves in water covered by brash ice

“…This paper continues the rigorous study (initiated in [3]) of a freely floating rigid bodies trapping time-harmonic waves in an inviscid, incompressible, heavy fluid, say water (see also [6,7,8] and [4]). We consider the infinitely deep water in irrotational motion bounded from above by a free surface unbounded in all horizontal directions, but unlike the cited papers dealing with the open surface, we assume here that it is totally covered with the brash ice.…”

“…Since W is a Lipschitz domain and ϕ ∈ H 1 loc (W ), it is natural to understand the problem, namely relations (2), ( 4) and (8), in the sense of the integral identity…”

“…The idea of the procedure is to seek bodies for a prescribed trapped mode. Subsequently, this approach was developed in [7,8], where various axisymmetric trapping bodies, motionless and heaving, were constructed as well as sets of multiple bodies some of which are motionless, whereas the others heave. In this section, we first apply the same idea to construct a family of motionless bodies trapping waves in the water covered by the brash ice and then consider the effect of the brash ice by comparing these bodies with those trapping waves in the open water.…”

Mean Value Properties of Harmonic Functions and Related Topics (a Survey)

“…The following consequence of Theorem 1.1 is not widely known, but has important applications in the linear theory of water waves; see the monograph [10], Sect. 4.1, where a proof of this assertion is given, and the article [11], where further references can be found.…”

On a cylinder freely floating in oblique waves

“…The original John's formulation of the floating body problem is rather cumbersome because he did not use matrices to express the equations of body's motion (the matrix form of these equations described below demonstrates their simple structure; see also [5], [6] and [7]). Anyway, the problem was neglected by researches during 60 years after publication of the article [8], in which the uniqueness theorem was proved for the threedimensional problem under the assumptions that the so-called John condition holds for the body (it is described below for two-dimensional geometry) and the frequency of oscillations is sufficiently large.…”