Roll of a Ship in Astern Seas—Metacentric Height Spectra

Abstract

Unstable roll motions can be a problem in astern seas because of fluctuations in the metacentric height. A formulation for the fluctuations in metacentric height is derived in terms of the distribution of instantaneous wave elevations and wave elevation accelerations along the ship's hull. This formulation is manipulated to derive an equation relating the spectrum of metacentric height fluctuations to the point spectrum of a long-crested, astern sea. Numerical simulations are performed to demonstrate the validity of the method. The singular behavior of the metacentric height spectrum at a frequency of g/4U has implications with respect to the safe operating speeds of a ship in random astern seas.