Keywords: Ship Wake, Synthetic Aperture Radar, ERS, SPOT

Abstract
In this paper, we give an outline of the current understanding of the imaging of ship wakes by a spaceborne SAR and by a spaceborne high-resolution optical sensor. We present examples drawn from the analysis of about 400 ship wakes in ERS SAR images and about 100 ship wakes in SPOT multispectral and panchromatic images. Our analysis shows that, consistent with current understanding of the imaging of ship wakes, the visibility and appearance of ship wake features depend on local parameters such as wind speed or sea state, heading direction of the ship relative to the looking angle (ERS SAR) or sun angle (SPOT). As a consequence, in ERS SAR images the Kelvin wake is visible in some form in only about 17% of cases, whereas the turbulent wake practically always shows up in both SPOT and ERS SAR images. We further discuss features which are not readily explained by current theory, such as Kelvin-wake signatures in ERS SAR images which are narrower than predicted, or dark lines forming a V-shape within the Kelvin-wake signature in SPOT images.

Introduction
A moving ship generates a wake which consists of a characteristic surface wave pattern confined to a wedge-shaped region behind the ship (Kelvin wake), and a turbulent wake along the track of the ship. Such a ship wake can often be seen in images acquired by the synthetic aperture radar (SAR) aboard the satellites ERS-1 and 2 or by the High Resolution Visible Scanner (HRV) aboard the SPOT satellites. The hydrodynamics of ship wakes is still an object of ongoing research as it is relevant for the construction of ships with minimal wave resistance, while thorough understanding of ship wakes in satellite images is relevant for correctly interpreting the images with the aim of monitoring ship traffic and compiling ship statistics.

Ship Wakes

Wave Pattern
The wave pattern generated on the water surface by a moving disturbance was first theoretically explained by Lord Kelvin, hence the name Kelvin wake. Without citing the details of the solution of the classical problem, we give its basic features for a point-like disturbance moving with constant speed V in a straight line on the surface of water of uniform depth D (Newman, 1977). For deep water, which in this context is defined by D>>2V²/g, where g is the acceleration of gravity, the wave pattern is confined to a wedge-shaped region behind the ship with a half angle b of 19.5° (the wake angle). Note that this angle is independent of the speed of the disturbance. The wave pattern consists of diverging and transverse waves, as sketched in Fig. 1. The propagation direction of the waves in the wake with respect to the heading of the disturbance is between 0° and 35.25° for the transverse waves, and between 35.25° and 90° for the divergent waves, while the wavelength decreases with increasing angle. The outward edge of the wedge, where transverse and divergent waves are superimposed to form so-called cusp waves, is usually the region of the wake with highest wave amplitudes, called cusp region or cusp lines.


Fig. 1: Schematic sketch of a ship wake (adapted from Hennings et al., 1999)
The qualitative features of this pattern are preserved if the point like disturbance is replaced by a disturbance of finite spatial extent which is more suitable to describe a real ship. However, the region around the ship and behind the ship up to a distance of several shiplengths (local wave disturbance region) shows a complex combination of breaking waves, bow and stern waves, very much depending on the speed, the shape and the propulsion system of the ship. The distribution of waves amplitudes behind the local disturbance region (free wave pattern region) depends on the ship as well. Thus, e.g., either the transverse or the diverging waves may dominate, and the cusp line may be more or less prominent. When the depth D approaches U² /g, the wedge widens: For D=2V²/g , the wake angle is b=20.5° and for D=1.2 V²/g, b>35° , eventually approaching 90° at the singularity. D= V²/g (Havelock, 1908). For shallow water which in this context is defined by D<2V²/g, the wave pattern is very different: There are no transverse waves, and no cusp lines. The divergent waves form a wedge whose half angle depends on the water depth and on the speed of the ship, sinb=(gh)^0.5/V.

For a swift ship with V=10 m/s (20 kn), V²/g=5.6 m, thus the deep water limit is valid when D exceeds about 10 m, which is generally the case for ship routes in the sea. The wavelength of the longest waves in the Kelvin wake is 64 m for V=10 m/s and 16 m for V=5 m/s.