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Frank Reed wrote:

> Now blow...
>
> Wind makes waves. The sea is no longer a flat reflecting mirror.
> Instead, the waves break the sea's surface into a multitude of facets,
> each like a small tilted mirror. A wave under the Moon towards the
> horizon has facets in the trough that are facing towards you and facets
> that are facing away. Part way down into the trough, there is a facet
> (like a small mirror) that sits at just the right angle so that a ray
> from the Moon is reflected towards the observer. The next wave after
> that also has such a facet, but it's a little further back in the trough
> of the wave. Wave upon wave out towards the horizon has a spot in its
> trough that is tilted just right so that the reflected ray is aimed at
> the observer's eye. Instead of just one image of the Moon (as in the
> case of the calm sea above), we see lots of little images (actually
> pieces of images) from all those little facets. And in every case, the
> geometry is just the same as in the case of a flat sea: angle of
> incidence = angle of reflection.
>
> Eventually, as you get towards the horizon, the reflected ray from the
> Moon that would head towards us by the surface in the wave trough out
> there would be blocked by the crest in front of it. There would be no
> way to make the angle of incidence equal the angle of reflection in a
> way that would beam the reflected ray towards the observer on the
> vessel's deck. At that point, you would no longer see a clear reflection
> of light from the Moon. And none of the waves after that would
> contribute to the light you see either.


Imagine an ocean surface between perfectly calm and a realistic sea,
this one having a perfectly sinusoidal swell running across it, away
from the Moon and towards the observer, but with no other imperfections
to mar its mirror-like surface. The steepest facets on such a surface
will occur precisely half way up the face of each swell, midway between
trough and crest. How steep they are will depend on the ratio of
wavelength to wave height in the train of swells. A slope of 1-in-5, or
11 degrees, might be realistic, though some swells will be steeper (and
others much less steep).

With the Moon's altitude at 45 degrees, the angle of incidence to these
steepest facets will be 56 degrees, meaning that the reflected ray will
leave the wave at an angle of 23 degrees above the horizontal (if I
haven't made any silly errors). Any angle  greater than about 5 degrees
should be ample to carry that ray clear above the crest of the next
swell in front. Hence, for sinusoidal swells, the cut off of the Moon's
path is not a matter of crests obstructed the reflected rays but of the
swell surfaces not providing sufficiently-inclined reflective facets.

A complex sea will, of course, provide some very steep (even vertical)
facets but it will also provide some of them high up the faces of the
waves, close to their crests. Moreover, a real sea is unlikely to be
running exactly down the path from Moon to observer, while other
directions of wave propagation will greatly change the geometry. All of
which makes the system too complex for such simple analyses as these.
However, I suspect that _some_ of the cut-off of the path of the Moon as
seen on the real ocean is due to wave crests obstructing the reflected
light (as Frank suggested) but that most of it results from a lack of
reflecting facets inclined at a sufficient angle to send rays towards
the observer's eye from points on the ocean's surface far away from
where the reflection would be seen given a mirror-smooth surface.


Trevor Kenchington


--
Trevor J. Kenchington PhD                         Gadus@iStar.ca
Gadus Associates,                                 Office(902) 889-9250
R.R.#1, Musquodoboit Harbour,                     Fax   (902) 889-9251
Nova Scotia  B0J 2L0, CANADA                      Home  (902) 889-3555

                     Science Serving the Fisheries
                      http://home.istar.ca/~gadus

   

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