result of wave interference between coherent sources replacing the hull of the vessel.

1. Short surface waves induced by the free-surface strain which is affected by ship-generated internal waves; the internal waves result from the interference of coherent sources replacing the hull (Keller & Munk 1970, Tulin & Miloh 1990, Miloh, Tulin & Zilman 1993).

2. Generation of short waves by incoherent point sources behind the ship (Munk et al. 1988); in the approach of Gu & Phillips (1995) incoherent sources simulate the oscillations of the edges of the turbulent wake.

In the present paper we investigate the attenuation of the short waves due to the presence of surfactant films compacted in the vicinity of the V-wake. Surfactants tend to concentrate at the free-surface and to alter the surface properties. In particular, this may result in a strong damping of ripples and short gravity waves (Levich 1962). It is common to interpret such a phenomenon as a Marangoni effect which is due to the gradient of the surface tension varying from point to point of the free-surface. The level of wave damping depends on many physical parameters of the water and the surfactants. The simplest mathematical model of the Marangoni phenomenon is based on the concept of a viscoelastic surface film and incorporates only three essential parameters: the water kinematic viscosity, v, the surface tension coefficient, σ, and the elasticity of the surface film, , (Levich 1962). There exist strong experimental evidence that the elasticity of the surfactant film depends on the concentration of surfactants (Pelinovsky and Talipova 1990) and can be as high as 30 dyne/cm for natural surfactants of Black Sea.

The characteristics features of the radar return such as the brightness of the V-arms and their extent can be expressed in terms of the radar back scatter cross-section. As was indicated by Peltzer et al. (1991) these characteristics may depend on the density of the surfactant films covering the sea surface.

The basic mechanism of radar backscattering. Herein we follow the work of Milgram (1988), where the radar back-scattering stems from the deformation of the initially flat free-su rface by a system of ship generated divergent waves with the wave length about 20– 30cm. This range is consistent with the L-band radar wave length and provides the Bragg resonance. However, as it was indicated in the work of Munk, Scully-Power & Zachariansen (1986) the radar picks up not only the particular magnitude of a wave number, but also the direction of wave propagation, i.e., the direction for which the wave crests are normal to the look of sight of the radar. Thus, the brightness and the extent of the two bright arms can be different. If the direction of the radar look of sight a provides the maximal available signal for the, say, right V-arm, the left V-arm may be practically invisible on SAR images. For instance, the experimental database of Brown (1985) includes 49 SAR images of different ships; 24% percents of them have wakes with two arms, 41% show one arm, while 35% of the images do not exhibit any bright envelopes. ¹ In all SAR images obtained by Shemdin (1990) the bright V-arms are visible only for those ships which travel in the same direction as the aircraft (α=0). Close scrutiny of SAR images presented in Shemdin (1990) shows that the extent, as well as the brightness of the arms are not the same. Moreover, in one of the reported images only one bright arm is visible.

The radar back scatter cross-section Θ depends on the wave elevation in the illuminated area, on the radar wave number k_r, the angle of incidence ψ, the length and width of the resolution cell 2l_c and 2b_c respectively and the complex bask scatter coefficient, C, depending on the particular type of radar. For radars with a horizontal polarization the back scatter coefficient can be expressed approximately as |C| cos² ψ.

According to the Bragg model the radar backscatter cross-section of a wavy surface per unit area is given by the following formula of Wright (1966, 1968):

(1)

Here the coordinates (ξ,η) pertain to a local coordinate system of a rectangular resolution cell co-planar with the radar line of sight. Thus, if the wave elevation (ξ,η) is known, the radar return also can be also computed.

Peculiarities of the ship wave-wake simulations. Simulation of the clean wave-wake of a ship is a classical problems of naval hydrodynamics and was considered by Kelvin (1891), Peters (1949), Ursell(1960, 1988), Wehausen & Laitone (1960), Sharma (1969), Newman (1970, 1971, 1987), Tuck, Collins & Wells (1971), Barnell