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

TL;DR

This paper analyzes how a floating body interacts with time-harmonic waves in water covered by brash ice, revealing frequency-dependent behaviors including wave trapping and trivial solutions at high frequencies.

Contribution

It introduces a spectral problem for coupled water-ice-body motion, identifying frequency regimes with finite energy solutions and constructing wave-trapping bodies.

Findings

Finite energy solutions exist near zero frequency.

Wave trapping bodies are constructed for certain frequencies.

Solutions are trivial at high frequencies.

Abstract

A mechanical system consisting of water covered by brash ice and a body freely floating near equilibrium is considered. The water occupies a half-space into which an infinitely long surface-piercing cylinder is immersed, thus allowing us to study two-dimensional modes of the coupled motion which is assumed to be of small amplitude. The corresponding linear setting for time-harmonic oscillations reduces to a spectral problem whose parameter is the frequency. A constant that characterises the brash ice divides the set of frequencies into two subsets and the results obtained for each of these subsets are essentially different. For frequencies belonging to a finite interval adjacent to zero, the total energy of motion is finite and the equipartition of energy holds for the whole system. For every frequency from this interval, a family of motionless bodies trapping waves is constructed by virtue of the semi-inverse procedure. For sufficiently large frequencies outside of this interval, all solutions of finite energy are trivial.