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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