Ac? ag? (2)
where Vs is the ship speed and g is the acceleration of gravity, con-
structively interfere to form wavefronts oriented at angles less than
v, = tant! V7 L°54:7°0, (3)
with respect to the ship track. These wavefronts are indicated by the
solid lines in Figure 6. Waves with wavelengths greater than ac form
wavefronts oriented at angles larger than Vc, as indicated by the dashed
lines in Figure 6. The longest of these are the stern waves, which have
wavefronts perpendicular to the ship track and propagate with a phase
velocity equal to the ship speed. Thus, their wavelength is given by
A = à (4)
As the longest waves, these are the most frequently resolved in SAR images.
Measurement of their wavelength allows the ship speed to be calculated from
Eq. (4).
The two sets of wavefronts shown in Figure 6 and discussed in the pre-
vious paragraph merge at the outer edges of the Kelvin wake pattern to form
the cusp waves. These waves have wavelength A. and wavefronts aligned at
an angle Vc, and are always located along a pair of lines oriented at
angles of
a, = ssin”! (1/5) = 19.5" (5)
with respect to the ship track. This pair of lines includes all of the
waves within the Kelvin wake and is known as the Kelvin envelope. Note
that the Kelvin envelope is not a wavefront, but only indicates the loca-
tion of the cusp waves. The cusp waves are usually the waves of largest
amplitude within the Kelvin wake, and therefore frequently appear in SAR
images. However, because of their shorter wavelength, they are often not
resolved, and thus appear as a bright line along the Kelvin envelope rather
than as individual wavefronts. Some examples of SAR data showing the
individual cusp waves do exist, however, as shown in Figure 7. This image
also shows the stern waves as well as the turbulent wake, discussed in the
following section.
In contrast with the case of ship-generated Bragg waves discussed
earlier in this section, the longer waves forming the Kelvin wake are only
imaged if there are ambient short-scale (Bragg) waves present on the water
surface. The Kelvin waves modulate the background return via the mecha-
nisms of tilt, hydrodynamic modulation, and velocity bunching discussed by
Alpers, et al. (1981). In particular, many cases in which the ship track
is nearly parallel to the SAR ground track (as in Figure 7), the stern
waves are apparently imaged via the velocity bunching mechanism. The nar-
row wakes formed from ship-generated Bragg waves are also occasionally
modulated by the stern waves, as can be seen through careful examination of
the narrow V-wake in Figure 2.
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