Re: phase vs group vel.

[Leigh Palmer <palmer@sfu.ca>]

> > Or a canoe. Another misconception which is rampant in the physics teaching
> > community is the idea that the Kelvin wake left by a ferry (or a canoe or
> > a duck!) has an opening angle that is somehow related to its speed and the
> > "speed of waves on the surface of water" in the same way that the opening
> > angle of the shock wave due to a supersonic object (a bullet or an airplane
> > or a car!) is. The phenomena are utterly unrelated, but still problems in
> > physics books are based upon this erroneous idea. It is easily demonstrated
> > that the Kelvin wake angle is independent of the speed of a canoe.
>
> It's a good thing that the Allied Military didn't know how wrong this was
> when they plotted the motion of convoys during WWII. And NASA is STILL
> laboring under this misconception by charting ships motion from satellite
> images! (The ship is too small to see but the wake is clear and is SAID to
> indicate the speed.)
>
> (But maybe your Kelvin wake isn't the same as these ship's bow waves???)

Nope. The bow wave is the same as the Kelvin wake. The bow wake can 
indicate changes in speed, but the angle for a uniformly moving boat 
is always 39 degrees. I think NASA knows about this. For example, see 
http://southport.jpl.nasa.gov/nrc/chapter4.html which contains a 
reference to "Wakes include both the invariant 39 degrees Kelvin wake, 
and in stratified water, occasionally an internal wave wake. I could 
not raise a more promising URL, http://daac.gsfc.nasa.gov/CAMPAIGN_DOCS/OCDST/shuttle_oceanography_web/oss_94.html
but perhaps you might try it later. The topic is not controversial, 
however. Bow wakes of all creatures great and small open (full angle) 
at 39 degrees.

Ah! I got it, finally. The above URL leads to:

       A moving vessel expends energy to overcome water resistance,
       creating eddies in the turbulent wake and generating a wave wake. Most of
       the power of a ship's engines is consumed in wave generation (Peregrine,
       1971). 
             The wake formed from the bow of a moving ship is a superposition
       of two families of waves. One family of waves, the longitudinal waves,
       propagates along parallel lateral lines. These lateral lines form an angle
       at the ship's bow of approximately 39 degrees. This angle (the
       Kelvin-wake angle) remains constant, depending neither on the speed nor
       on the shape of the vessel in deep water. It is determined by the principle
       that the group velocity of waves in deep water is equal to one-half the
       phase velocity. 
             The second family of waves, the transverse waves, forms immediately
       behind the ship and at a right angle to the direction the ship
       is traveling. Both families of waves are stable relative to the ship, but
       the transverse waves are modified by the turbulent wake behind the ship.
       There, the stern wake is modified by a large amount of foam and bubbles
       created when waves break at the ship's bow and by cavitation from the
       propellers. 
             Another interesting phenomenon is the narrow V-shaped bow waves
       imaged by the SEASAT synthetic aperture radar (SAR). Various analyses
       have defined the backscattering conditions that produced the appearance of
       a less-than-39-degree bow wake when a moving ship is remotely sensed
       in microwave frequencies (Vesecky and Stewart, 1982). 
             From visual observations and photography, astronauts aboard the
       space shuttle provided information on ship wakes that differed from that
       obtained from a spaceborne SAR. Stern wakes extending behind a ship for
       more than 200 kilometers were observed routinely. Perhaps the most
       surprising, and intriguing, however, were the "nested-V" bow waves
       photographed during shuttle missions in 1984 and 1985 (Munk et al.,
       1987). Their geometry differs completely from those imaged by a SAR,
       requiring a new theoretical approach to the response of ship wakes to the
       local sea. 

The Kelvin wake, however, is that most conspicuous v-shaped bow wake (and 
stern wake) with which we all *should* be familiar. But, as I said earlier, 
this is a widespread misconception among physics teachers.

Leigh