Radar Backscatter of a Vlike Ship Wake from a Sea Surface Covered by Surfactants
G.Zilman, T.Miloh (TelAviv University, Israel)
ABSTRACT
In calm water synthetic aperture radar (SAR) ship wake images frequently form of a bright V with a narrow half angle of 2–3 deg. These images persist for many kilometers behind the ship and can be explained as a result of the Bragg scattering from shipgenerated divergent waves (Milgram 1988). The present work is concerned with the generation of short divergent gravity waves and their radiation from the hull of a displacement ship moving on a freesurface covered by surfactants. Here it is shown that a contaminated freesurface significantly reduce the radar back scatter crosssection and drastically influence the Vwake detectability. Explicit formulas for the radar back scatter crosssection are obtained and numerical examples for their use are presented. The theoretical results are sown to be in a good qualitative and quantitative agreement with the experimental results of Shemdin (1990).
1
Introduction
Background. A moving ship generates a distinctive Kelvin freesurface waves system. It consists of two types of waves, transverse and divergent, which are located between two lines with a half angle of 19.5°. However, airborne synthetic aperture radar (SAR) images of the sea surface frequently display quite a different noticeable wave pattern in a form of a narrow Vlike angle of 2°–3° degrees (Munk, ScullyPower & Zachariasen 1986, Lyden et al. 1988, Shemdin 1990). Such wakes exhibits a central “dark area” and two ”bright white arms” existing as far as 6–8 km behind surface ships.

Dark Vwake visible only in SAR images;

Two confining bright arms;

Optically visible Kelvin waves which are usually invisible in SAR images.
There is a believe that the bright arms of the Vwake stem from the radar backscattering of short surface waves which are generated by the ship hull or its turbulent wake. Such a hypothesis is based on the Bragg scattering mechanism, which implies that a radar, roughly speaking, picks up a single wave number. The typical wavelength selected by Lband radars varies somewhat between 25–30 cm. Thus, it is plausible that such waves are presented in the ship wake.
To explain the Vwake phenomenon so far three alternative and somewhat mutually incompatible mechanisms have been invoked. They relate to:

Short divergent waves generated by a ship moving with a constant speed (Milgram 1988); in this approach the wave wake is a

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

Short surface waves induced by the freesurface strain which is affected by shipgenerated 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).

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 Vwake. Surfactants tend to concentrate at the freesurface 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 freesurface. 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 Varms and their extent can be expressed in terms of the radar back scatter crosssection. 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 backscattering stems from the deformation of the initially flat freesu rface by a system of ship generated divergent waves with the wave length about 20– 30cm. This range is consistent with the Lband radar wave length and provides the Bragg resonance. However, as it was indicated in the work of Munk, ScullyPower & 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 Varm, the left Varm 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 Varms 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 crosssection 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 cos^{2}ψ.
According to the Bragg model the radar backscatter crosssection 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 coplanar 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 wavewake simulations. Simulation of the clean wavewake 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
^{1 } 
The authors are gratefull to Dr. P.Wang for this information. 
& Noblesse (1986), Barr & Price (1988), Noblesse, Hendrix & Barnell (1989), Nobless & Hendrix (1991), Ponizy et al. (1994), Nakos & Sclavounos (1994). The effect of viscosity on ship waves has been investigated, for example, by Wu & Messick (1958), Cumberbatch (1965) and Kinoshita (1981). The effect of the surface tension on ship waves was investigated by Crapper (1964) and Sharma (1969). The unsteady Kelvin wake was investigated by Mei & Naciri (1991), Eggers & Schultz (1992) and Cao, Schultz and Beck (1992).
The striking feature of the numerical simulations implemented by Noblesse and Hendrix (1991) is that they ..”predict short divergent waves too steep to exist in reality within a significant sector in the vicinity of the ship track ”. This difficulty was attributed by Milgram (1988) to a collapse of the divergent waves. In the framework of the present formulation such a pure theoretical obstacle can be lifted in a natural way since all short and nonrealistic steep waves are completely damped out by a surfactant film.
The Vwake persist for many kilometers. Obviously, such a distance is much larger then the typical ship length. Thus, one can employ an asymptotic far field estimate for the wave elevations which greatly alleviate the consequent analysis. However, even a simplified expression for the wave elevation still makes the integration in the formula (1) not easy to perform. The relevant computations usually are time consuming and quite demanding of computer resources. The reason being that the expressions for the wave elevation are represented by integrals which hardy can be evaluated in an explicit form, and the numerical computation of (1) may involve millions of grid points.
It was noted by Milgram (1988), that in the far field the waves generated by a ship behaves almost as they came from a single location. In other words, far behind a ship the fine details of its shape may be inessential. It allows us to use a simplified description of the ship hull which takes into account only such gross geometrical parameters as ship length/beam ratio and the fullness coefficient of the waterline α_{W}. Such an approach provides an opportunity to select from the variety of ship forms only those which allows us to obtain an analytic representation of the wave elevation in the far field. Once such an analytic expression is known, one can try to find an asymptotic expression for the radar back scatter crosssection Θ.
The plan of the present work. Section 2 of this paper provides the mathematical outline for evaluating the wave elevations. It is based on linear water wave theory. In Section 3 we present the derivation of the Green function for a source moving below the freesurface covered by insoluble surfactant and an asymptotic expression for wave elevations. In this section we also discuss the damping effect of the surfactant film. An analytic expression for the radar back scatter crosssection is derived in Section 4. Starting from Section 5 the theoretical results are compared against the experimental data of Shemdin (1988). The particular numerical example involves a tanker Bay Ridge with the following characteristics:
Table 1. The particulars of the tanker Bay Ridge.
Length, m 
334.9 
Bean, m 
43.7 
Draft, m 
21.4 
Speed, m/s 
7.5 
The characteristics of the radar with received horizontal polarization which was used in the experiment of Shemdin are represented in Table 2:
Table 2. Aircraft LBand SAR parameters.
Wavelength, m 
0.246 
Nominal resolution, m×m 
11×11 
Ground range resolution (at incident angle 30°),m 
16×16 
Ground range resolution (at incident angle 50°),m 
10×10 
2
Wave elevations induced by a moving ship.
Let us select the rectangular axes of the coordinate system attached to a moving ship in such a manner that the x and yaxes are situated on the undisturbed freesurface, and the zaxis is directed upward. We define the surface S of the hull by the function y=±f(x,z). Our goal is to determine the freesurface profile =(x,y) generated by a ship moving with constant velocity U in the direction of the positive xaxis. In the frameworks of the linear analysis we represent the resulting wave elevation (x,y) induced by the ship as a sum of wave elevations due to sources of intensity q_{h}(x,y,z) distributed over the surface S and wave elevations which stem from the sources q_{w}(x,y) distributed over the contour of the waterline l_{w} (Brard 1972):
(2)
where
(3)
and
(4)
Here N(x_{0},y_{0},z_{0}) is a point on the surface S and M(x,y,z) is a field point · ζ_{h}=G(M,N) and ζ_{w}=ζ_{w}(M,N) are two related Green functions. In general, for irrotational fluid motion the source density q_{h}(x,y,z) can be found as the numerical solution of a corresponding boundary integral equation. The relation between the strength of the surface and the line source distribution is:
where n_{x}=cos(n,x) is the projection of the normal to the ship surface on the xaxis, and g is the acceleration of gravity.
In the present work we consider a viscous fluid, but assume that the source density still can be calculated by invoking the potential theory. The combined effect of the fluid viscosity and the surfactant film elasticity will be included in the Green function. Such a heuristics approach has a real physical background. For the potential flow the source density distribution displays the form of the ship whereas the Green function manifests the physical properties of the fluid and the specific kinematic and dynamic boundary conditions to be applied on the freesurface. This approach can be partially improved by accounting for the thickness of ship boundary layer. Based on the work of Lavretntiev (1951) Kinoshita (1980) has been demonstrated that the Michell source distribution with the effect of the boundary layer displacement thickness is almost identical to the source distribution without a viscous effect, except a particular range near the stern. Havelock (1935) demonstrated many decades ago that the influence of the viscosity on the near stern flow can be accounted for by introducing a proper small “deformation” of the stern. It has been mentioned above that far behind a ship the fine details of its surface may be inessential. Thus, we assume that for estimating the far field wave wake the source density can be calculated on the basis of potential theory according to the formula:
(6)
where γ(x,y,z) is a function which lamps the specific features of the shape of a ship. For a thin Michell type ship γ(x,y,z)=2, and the integration in the formulas (3) and (4) can be performed at the ship the centerplane. For a ship of finite width γ varies from point to point of the ship surface, but it is bounded, i.e., 1<γ <1.5 (Kostyukov 1968). In order to simplify the analysis and to obtain analytical results we employ the relation (6) with a constant coefficient γ 1, but perform the integration on both sides of the surface S. Such an approach has been previously justified by Milgram [20] who noted that ”comparative calculations of divergent waves and their scattering crosssection using thin ship source distribution (distributed on the actual ship shapenot the centerplane), and the source distribution obtained from its linear theory integral equation, show differences in details, but not in general form”.
3
The Green function of a moving surface disturbance.
3.1
Governing equations.
We consider a disturbance (a normal stress P_{n}) concentrated over a infinitesimally small area in the vicinity of the origin O of the coordinate system Oxyz. The disturbance moves rectilinearly with constant velocity U on the freesurface ζ_{p}=ζ_{p}(x,y) of an incompressible viscous water covered by a surfactant film. A strong similarity between the wave elevation induced by a moving source and the wave elevation induced by a moving impulse of a normal stress applied to the freesurface is well known. For an inviscid fluid and irrotational fluid motion the wave elevation induced by a pressure impulse P_{n} moving over the freesurface is:
Once the function ζ_{p} is defined, the functions G_{h,w} can be reconstructed by using a formal substitution:
P_{n}=2iU sin θ. (7)
The linearized (Oseen) equation of fluid motion is expressed in the moving coordinate system as:
(8)
divV=0, (9)
where V is the vector of fluid velocity, p is the pressure in the fluid, ρ is the water density and v is the kinematic viscosity. This equation can be splited into potential and rotational parts
V=grad +V_{R}(u,v,w),
such that
∇^{2}=0, (10)
and
(11)
divV_{R}=0. (12)
3.2
Boundary conditions.
The linearized kinematic condition on the freesurface is:
(13)
In the presence of surface tension the linearized normal stress dynamics boundary condition on the freesurface can be written as:
(14)
where µ is the water dynamic viscosity and σ is the surface tension coefficient. In general σ depends on the surfactant concentration Γ which, in turn, is a function of the coordinates (x,y) and thus, σ=σ[Γ(x,y)]. We assume that the concentration Γ of a surfaceactive agent is Γ=Γ_{0}+Γ′, where Γ_{0} is the constant concentration on the undeformed surface and Γ′≪Γ_{0}. For slightly viscous fluid, low fluid velocity and smallamplitude waves it is common to assume that in equation (14) the surface tension coefficient is constant, σ=σ(Γ_{0}). However, the gradient of the surface tension affects the tangential force
and the tangential shear stress conditions on the freesurface become:
(15)
(16)
where p_{tx} and p_{ty} are the x and ycomponents of the shearing stress. Here the shearing stress vector can be expressed through the surfactant film concentration Γ(x,y) and the concentration Γ_{0} as (Levich 1962):
where is the elasticity of the surfactant film
and
We assume that the insoluble surfaceactive agent is fully swept along with the liquid and that the influence of diffusion on the concentration distribution of the surface active agent may be disregarded. For low fluid velocity and smallamplitude waves the equation for the conservation of matter in the case of mass flow on an almost planar freesurface yields:
(17)
3.3
Fourier transformation of the boundaryvalue problem
Let us denote the Fourier transform (FT) with respect to x and y as:
(18)
The Fourier transform of (10) gives:
(19)
with a solution
(20)
where The FT of (11) yields:
with a solution
where
(21)
and Re(l)>0. Applying FT to (12) and (14) we have:
(22)
where T=σ/ρ. From (15) and (15) it follows:
(23)
(24)
Finally, equations (12) and (17) yield:
(25)
(26)
The next step is to find the Fourier transform and its inverse ζ_{p}. This can be done by solving a system of linear algebraic equations with respect to unknown parameters A, C_{1}, C_{2}, C_{3}. After some simple, but somewhat tedious calculations we obtain the following expression for
where
D=–k(ω^{2}–βmk^{2})(s–2viωl)+ ω(2vkm+iω)[2vωk^{2}+i(βlk^{2}–ω^{2})]
β=/ρ, m=l–k, ω=Uk_{x} and s=g+Tk^{2}. Performing the integration in the polar system of coordinates, taking into account (7), neglecting in the numerator a small term incorporating β ≪1 and small terms of order v^{2} in the denominator, we obtain the following expression for the regular part of the Green function:
where
(27)
Ω^{±}=(x sin θ±y cos θ)
E(k,θ)=(kτ^{2}–s)(τ^{2}–βl)+2ivkτ^{2},
τ=U sin θ,
and
(28)
Here and in the sequel it is understood that in all complex expressions for the wave elevation similar to (27) have to be considered only the real part.
3.4
Poles of the integrand.
In the far field the asymptotic behavior of the Green function is determined through the poles of the integrand, or, in fact, by the roots of the equation:
E(k,θ)=0. (29)
The effect of the inclusion of viscosity is to move the poles off the real axis, and viscous damping is thereby obtained. In the case of an inviscid fluid the two corresponding roots are real and can be found explicitly as:
(30)
They corresponds to the capillary, capillarygravity and gravity waves regions. The upper sign in this expression pertains to short capillary waves which are not important for our analysis for two reasons. First of all, they decay very fast with increasing distance from the disturbance and, secondly, they do not resonant with the radar wave length λ_{r} ~ 20–30cm. Thus, in the sequel we will
be concerned with short gravitycapillary waves which are defined by the lower sign of this relation. It is important to note that far downstream from the disturbance the integration in formula (27) is essential only for such values of θ which satisfies the inequality:
(31)
For a viscous fluid with contaminated freesurface the roots of (30) become complex and the yielding poles can not be solved exactly. Only an approximate location of the poles can be obtained. To a good approximation the real part of the roots obeys the relation (30) while the imaginary part may depend on the elasticity of the surfactant film significantly. Thus, the complex roots may be written as follows:
k_{c}=k_{0}(θ)–iδ(θ), (32)
where δ>0 incorporates the damping effects of the water viscosity and the surfactant elasticity. If the initial value of the real root k_{0}(θ) is known, then the perturbation analysis of the of the dispersion relation (29) with respect to the small parameter δ/k_{0}≪1 yields an algebraic expression for the function δ=δ[k_{0}(θ),θ,v,]. The additional complication for such an analysis arises from the branch points of the function E which are imposed by the relation (28). In order to avoid this mathematical difficulty we assume that
Actually, since our interest resides in the range of wave numbers k ~ 25 m^{–1}, ship velocity U ~ 10 m/s, water kinematic viscosity v=10^{–6} m^{2}/s and θ satisfying the inequality (31), it follows that k^{2}≪ikU sin θ/v. In the limiting case of a viscous noncontaminated water the damping coefficient can be written as:
δ^{0}=–4g^{2}v/U^{5} sin^{5}θ. (33)
The ratio γ=δ/δ_{0} characterizes the damping amplification factor which stems from the presence of surfactant film. It is plotted in Fig.2.
It can be seen clearly that the surfactant film greatly intensify the wave damping for small θ angles.
3.5
The Kochin function.
According to (3) and (4) the wave elevation induced by sources distribution over the hull surface S and the waterline contour l_{w} can be written as follows:
(34)
(35)
where
and
(36)
represent the Kochin functions. For short divergent waves with the wave length λ_{w} ~ 20–30cm the hull integral is significant only over a small depth of order λ_{w}. In this case H_{h}(x,y,z) H_{h}(x,y,0) and q_{h}(x_{0},y_{0},z_{0})q_{h}(x_{0},y_{0},–0). In
the vicinity of the waterline a ship has almost a vertical board and thus:
(37)
where
It can be demonstrated that for short divergent waves the which implies that the contribution of the line integral is dominant in the final expression for the wave elevation.
3.6
Simplified waterline
As it was earlier mentioned, the fine details of hull shape are not really essential as far as the far wave field is concerned. Thus, we choose a particular polygonal contour of ship waterline as:
where L is ship length, (L_{0}/L) ~ 0.6–0.7 and
Straightforward calculations give the following expression for the wave elevations:
(38)
where
and
3.7
Asymptotic expression for the wave elevation.
In order to determine the asymptotic behavior of (38) well downstream let us consider some ncomponent of the integrand tacitly omitting the subscript n:
(39)
Following Whitham (1974) let us represent the integral with respect to θ as a sum of two integrals:
In the first of them the function sin(θ–χ) is positive whereas in second one it is negative. We deform the contour of integration with respect to the complex variable k and represent the corresponding integral as a sum of three integrals: an integral along the real halfaxis, an integral along the positive (negative) imaginary halfaxis and a semicircle of an infinitely large radius connecting the ends of the real and imaginary axes. The roots of the equation E(k,θ)=0 with respect to k are located in the lower halfplane. Thus, if sin(θ–χ)>0 the contour of integration is directed along the negative imaginary halfaxis and the poles of the integrand give a contribution; if sin(θ–χ)<0 the contour of integration is directed along the positive imaginary halfaxis and the poles of the integrand do not give a contribution. This also insures that the contour integral along the semicircle vanishes. For the purpose of estimating the order of magnitude of the integrals along the imaginary axes their limiting values as R → ∞ may be taken in the first approximation for a clean inviscid water without surface tension. It has been shown by Ursell (1960) that for R≫1 they are of the order of R^{–3} and thus can be neglected. We may conclude therefore that for large values of R formula (39) can be written as:
(40)
where
Δ(θ)=exp[δR sin(θ–χ)],
and
Φ(θ)=k_{0}(θ)sin(θ–χ).
Similar analysis can be performed for the integrand of (38) incorporating the terms sin(θ+χ). However, as it will be shown below, within the intended accuracy such integrals do not contribute to the final result.
Further calculations are based on the stationary phase method. For large values of R the integral (40) can be estimated as follows:
where θ_{s} denotes the root of the equation
Φ′(θ)=0.
We intend to find an analytic expression for the wave elevation in the vicinity of a small angle χ which is comparable to the magnitude of the Vwake angle χV. Thus, it can be assumed that sin(θ±χ) ≃ θ±χ. Furthermore, since χV≫ 4gT/U^{4}, the root of (30) can be estimated as follows:
which yields
It is possible now to obtain a simple expressions for the only point of the stationary phase θ_{s}
θ_{s}=2χ,
which define the function Φ(θ_{s}) and its second derivative Φ″(θ_{s}):
(41)
Hence the final expression for the wave elevation can be written as:
(42)
4
Asymptotic expression for the radar back scatter crosssection.
Consider an orthogonal local coordinate system O_{1}ξη with an origin located at the center of the resolution cell. The coordinates (ξ,η) are rotated by an angle α in the counter clockwise direction from (x,y) coordinates:
x=–X+ξ cos α+η sin α,
y=Y+η cos α–ξ sin α.
Here (–X<0,Y>0) are the coordinates of the center of the resolution cell in the coordinate system Oxyz. Substituting (42) into (1) we have:
(43)
where
(44)
and k_{e}=2k_{r} sin ψ. The function _{n}(R_{n},η) for R_{n}≫1 has a distinctive extremum in some points It allows us to employ the idea of the stationary phase method, but with some modification since the phase function by itself depends on a large parameter. We expand next the function _{n} in Taylor series
(45)
in the vicinity of some point η^{0}≤b_{c} such that In order to simplify the final results we assume that X≫1, cos α=1 and sin α= α. After some tedious calculations the following expressions can be obtained:
(46)
(47)
where
For U ~ 10m/s, k_{r} ≈ 25m^{–1}, sin ψ ~ 1/2 and X ~ 1,000m the numerical estimates of the derivatives show that and . Thus, the higher derivatives in the expansion (45) can be disregarded but the second derivative should be kept.
In the expression (46) the coordinates of the resolution cell are the geometrical parameters of the problem which can be chosen in order to provide the maximal back scatter crosssection. Let us further assume that
where j is some particular integers (i=1,…,8 ). In this case the coordinates of the center of the resolution cell are situated on the line
(48)
where Thus, we can define the Vwake angle as:
(49)
Consequently, the point of stationary phase is:
The substitution of (48) into (47) gives the following expression of the phase function:
Finally, the dependence of the function (44) on the coordinates of the local system is rather weak, and thus it can be estimated as:
G_{n}(R_{n},χ_{n})~G_{n}(R_{0},χv),
where
Now the integration in the formula (43) can be performed analytically which yields the following expression for the radar back scatter crosssection:
Θ=A_{1}A_{2}A_{3}A_{4}A_{5}, (50)
where
(51)
(52)
(53)
(54)
and
A_{5}=Δ^{2}(2χv). (55)
Here
and
(56)
Each factor in the product (50) allows a clear physical interpretation: A_{1} reflects the characteristics of the radar, A_{2} the parameters of the ship, A_{3} the direction of propagation of the divergent waves and the radar look of sight, A_{4} the interference between waves and their radiation decay and
A_{5} the radar back scatter crosssection attenuation affected by the fluid viscosity and surfactant film elasticity. It is interesting to note that for a ship with a length, say, L ~ 200m and width of a resolution cell about 2b_{c} ~10m, at least one point of the stationary phase (n=1,…,8) is equal to zero whereas the rest of them are situated outside the resolution cell. Thus, the contribution from the rest of these points in the formula (54) is negligible. That leads to an additional simplification of (54):
(57)
where C(ρ_{0}) and S(ρ_{0}) are the Fresnel integrals and
(58)
It is seen now that for ρ_{0}≫1 the asymptotic estimates of Fresnel integral gives: C(ρ_{0})=S(ρ_{0}) ~ 1/2 which means that the brightness of the signal does not decay along the bright arms. In practice the value of ρ_{0} is finite, but still can be large enough to provide a strong baskscattering signal. In fact, if ρ_{0}<1.5 the function A_{4} behaves as 1/X whereas for larger values of ρ_{0} it is an oscillatory function with a slowly decaying envelope.
5
Numerical simulation.
It is common to represent the radar crosssection in terms of its nondimensional rise above the background radar back scatter crosssection Θ_{0} which depends on the sea state. The value Θ_{0} can be estimated by a simple formula of Wright (1968). However, herein for the sake of consistency we prefer to invoke the experimental data of Shemdin (1990) which are represented in Table 3.
Table 3. Radar crosssection Θ_{0}for different wind speed (ψ=30°).
Sea state 
Wind speed, m/s 
Θ_{0}(dB) 
1 
1–2 
–30 
2 
2–4 
–22 
3 
4–6 
–17 
4 
6–8 
–13 
The dependence of the nondimensional rise of the radar back scatter crosssection above the background =Θ/Θ_{0} for Sea State 1 is shown in Fig.3.
According to the experimental data of Shemdin the radar cross section decay along bright arms exhibits distinctive extrema, which, apparently, can not be explained by the noise of measurements. Moreover, the last significant maximum of the decaying curve occurs at the distance X about 3–4 km aft of the ship. According to the presented theory the various extrema result from the wave interference reflected by the relation (54). Noticeable, that according to the experimental data the length of the Vlike bright arms varies between 5–6 km. It is reasonable to define the length of the bright arms L_{V} as a coordinate X_{V} where Θ_{0}<1.0 dB. Under such a definition the theoretical prediction gives the value of L_{V} ~ 4.5–6.0 km which is a good agreement with the experiment.
The bright Vwake arms appear in the SAR images only under the conditions of light wind and small wave height. The theoretical results presented in Fig.4 are in a good agreement with these experimental observations.
It is seen that for Sea State 2 the visibility of the bright arms is actually much less then for the Sea State 1 but still noticeable. For Sea State 3 the visibility of the bright arms is doubtful. For Sea State 4 the magnitude of the radar crosssection is below the radar crosssection of the background level and thus can not be observed.
In Fig.5 it is demonstrated the dependence of the radar crosssection on the elasticity of surfactant film.
It is seen that even a relatively small surfactant elasticity (about 10 dyne/cm) significantly reduce the radar crosssection and the length of the Vwake. For relatively large but plausible values of surfactant elasticity about 25 dyne/cm the Vwake can not be anymore detected.
6
Summary.
An important progress has been achieved toward improving our general understanding of intriguing phenomenon of ship generated narrow Vwakes. The proposed analytic solution of the problem incorporates the important effect of sea surface contamination. It has been shown that the radar crosssection is a decaying oscillatory function along the bright Varms. The asymptotic behavior of radar crosssection for a large distance X aft of a ship is governed by the asymptotic behavior of the Fresnel integrals C(ρ_{0}), S(ρ_{0}) where ρ_{0} is defined by (58) and (49).
The theory predicts a slow decay of the radar crosssection with increasing distance X until the value of the parameter ρ_{0} exceeds some threshold value ρ_{thresh}1.5. When ρ_{0} is below this critical value, the radar crosssection decays along the bright Varms as 1/X.
It has been demonstrated also that even a relatively small contamination of sea surface can change the radar back scatter crosssection drastically. According to the results of the present study the distance behind the ship for the surfactant film with elasticity ~ 20 dyne/cm to diminish the radar back scatter crosssection by a factor e^{–1} can be only few hundreds meters.
In this context of the present research it has to be mentioned that the visibility of the dark turbulent wake in Sea States 1–4 presumably can be explained by the attenuation of windgenerated waves due to the contamination of sea surface within the Vwake (Peltzer et al. 1991). The physical mechanism of compacting surfactant films in the immediate vicinity of the Vwake is an important followup to this investigation.
7
Acknowledgment
The authors would like to express their gratitude to Dr. L.Shemer for many valuable advises. Dr. E.Kit made interesting comments influencing the research. Dr. E.Pelinovsky during his visit to TelAviv University read the draft of the manuscript and his critical comments are appreciated by the authors.
The research was partially supported by the Israeli Ministry of Science.
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