Palm distributions of wave characteristics in encountering seas

TL;DR

This paper derives Palm distributions for ocean wave characteristics, such as wave slope and height, for waves observed along a line and encountering a ship, with a physical interpretation involving Doppler shift.

Contribution

It introduces new Palm distribution formulas for wave steepness in two different encounter scenarios and explains their relation through Doppler shift effects.

Findings

Derived explicit Palm distributions for wave characteristics.

Established the relation between distributions for fixed line observations and ship encounters.

Provided physical interpretation involving Doppler shift.

Abstract

Distributions of wave characteristics of ocean waves, such as wave slope, waveheight or wavelength, are an important tool in a variety of oceanographic applications such as safety of ocean structures or in the study of ship stability, as will be the focus in this paper. We derive Palm distributions of several wave characteristics that can be related to steepness of waves for two different cases, namely for waves observed along a line at a fixed time point and for waves encountering a ship sailing on the ocean. The relation between the distributions obtained in the two cases is also given physical interpretation in terms of a ``Doppler shift'' that is related to the velocity of the ship and the velocities of the individual waves.