This morning, while waiting for the sun to come up enough to start skiing, it struck me that I didn’t really know the reason why the reflection of far-off lights on a ripply lake is mostly in a column, or on a line between me and the light.
As a first-pass approximation, if the lake is ripply, it seems like the reflection of the light would be spread out in a circle around the light. Can someone explain this to me?
The reflection of a far-off light can vary, depending upon the shape of the water’s surface. If it’s perfectly flat, for instance, it acts like a mirror, and the reflection of a point source is a point. You need waves to get that “pillar” effect.
In any event, there are quite a lot of different effects you can get. There’s a section on thios is my favorite oddball physics reference, Jearl D. Walker’s The Flying Circus of Physics. Check it out, along with its references. I’m not sure, but there might be a section on it in M. Minnaert’s book The Nature of Light and Color in the Open Air, which is still in print.
That said, I don’t have a quick answer to your question. It seems to me that small ripples perpendicular to a line between you and the light ought to give you a straight “pillar”, but I’m not sure about other cases.
The reflection of far-off lights does radiate in a circle. Mostly that light isn’t headed your direction.
This is a matter of a single, simple rule: You can see any reflection where your eye sees at the same angle to the reflecting surface, as the surface is related to the lighted object.
If the lake was completely smooth, you’d only see one reflection.
But when there are ripples, from time to time the angle of a ripple may briefly match the angle needed to reflect the light at you.
When the ripple is very close to the place where a flat lake water surface would reflect, then you see the reflection frequently. That is, the water is frequently close to the natural angle of reflection of the flat surface.
Ripples that are far out of line with you and the light source only very occasionally get to the angle of the natural, flat reflection, so you rarely see light reflected from them. Eventually there are ripples that even at their most extreme, don’t match the angle necessary to reflect the light at you.
Water reflections appear in a column, if this helps, that match the overall direction of the light as it travels to your eye. Another way to consider this is to imagine how weird it would be if a ripple far off to one side was reflecting at you. You’d wonder what was coming out of the surface, right? You don’t expect water that’s 45 degrees to one side to reflect anything, unless it’s tilted upward for some unusual reason.
That explains most of the effect. In regard to lesser considerations, even relatively perfect surfaces such as a calm lake don’t reflect light perfectly, because some of the light gets “scattered”. That’s a way of saying there are microscopic imperfections in the surface. Also, the air between you and the object itself scatters light. The longer and more hazy the distance, the more gets scattered.
Okay, after scribbling lots of diagrams on my whiteboard, I think I have the correct picture in my head. Lets see if I can get it out in words, or if the dryerase fumes will interfere. 
There are (at least) two things going on. One is that the surface of the lake is rough because of the waves. Another is that you are looking at the lake at a low angle.
If the lake was perfectly smooth (like a mirror) then you would see, not a column, but a single image of the light, located at the point where the angle of incidence of light from the lamp onto the perfectly smooth horizontal surface equals the angle of reflectance to you, the viewer.
Because the lake is rough, though, the lake’s surface isn’t perfectly smooth. There are patches of lake surface at all angles, so you don’t see the image of the lamp at just at one place, but instead see many partial images of the lamp from many different places on the lake’s surface.
Imagine that instead of a lake, you had a vast field of little mirrors whose position you could remotely control, and you want to line them all up so that each one gives you an image of the lamp. There’s one mirror that will be excatly horizontal, kinda near the center of the “lake.” (Assuming that the light is about as high off the ground as your head.) Now, think about how you’d have to angle the other mirrors to catch the lamp’s reflection.
All the mirrors along the line between you and the lamp would just be tilted toward or away from you, not to the left or right. But mirrors off to the side need to be tilted left-right, and the amount that they need to tip increases rapidly with distance from the line of sight.
Now, think about the way that waves look from your perspective on the shore. No matter what the actual orientation of the waves, as you look across the lake from a low angle, the waves tend to look stretched out left-to-right, right? This means that most of the surfaces you can see have a toward-away (i.e. parallel to the line of sight) tilt, not a left-right tilt, ergo, you only see reflections along the line of sight to the lamp, or, a “column” of light.
That last paragraph got a little dodgy. Maybe someone can shore up (ha ha) my reasoning a bit.
I should add: if you were looking straight down on the lake, you would see a circular reflection from the light.
You may notice this exact effect when you look at a streetlight through the leafless branches of a tree whose bark is wet (or even better, if there’s been an ice storm and the branches are coated with a thin layer of ice). The branches are randomly oriented (more or less) but only branches perpendicular to a ray of light reflect the light toward you, so the reflected light forms sort of a halo around the lamp.
Pokadyne, thanks for taking the time, but I still don’t get it. By the way, I found an excellent photo on the web showing this effect. In the photo, notice that the pillar of reflection is very long, but is very narrow. There are almost no “little mirrors” tilted to the side to reflect a light to me, but they seem to be free to tilt forward and backwards to a great degree.
As for the waves explanation, these aren’t really long waves, but just a little roughness in the surface of the water, caused by a breeze. I’m not sure that matters.
I’ll play around by putting some numbers to your little mirrors analogy and see if I can get anywhere.
The idea is that in fact the little mirrors are not free to tilt very much in any direction. Roughly speaking, for deviations to “left” and “right”, the required tilt depends on the ratio of the distance y from the “ideal-reflection point” (where the image would appear if the lake were smooth) to your height h above the lake surface. For example, for you to see the reflection from a point with y = h, the lake surface has to be at a 45° angle.
For deviations forward and backward, though, the tilt depends on the ratio of the distance from the ideal-reflection point to the horizontal distance R between you and the ideal-reflection point. Actually, when you work it out, it’s even smaller than this; there’s an extra factor of h/R in there, so the required lake surface angle is something like xh/R[sup]2[/sup].
When the reflection off the surface is at a very low angle (in the picture, since the Moon is about 0.5° wide, it looks like the Moon is about 2° above the horizon), your horizontal distance to the ideal-reflection point is much larger than your height (in this case, about 30 times as large). This works out (if the lake is pretty smooth) to a factor of about 30[sup]2[/sup]=900 between the left-right and forward-backward extents of the reflection.
CurtC, if the light was directly behind your head, and you were looking straight down into rippled water, the light would reflect in a rough circle, just as you say.
But when you aren’t even close to looking straight down, you’re seeing all sorts of reflections catching the light at various points as it’s travelling along the line from the distant shore lights to your eye.
The effect you see in the photograph you’ve cited is because the lake has thousands of shallow ripples, each frequently forming a mirror pointed to your eyes. For the sun to be reflected from a wave, say at the right side of the picture, the wave would need to be raised from the water nearly vertical.
Imagine yourself at the edge of the photo with a mirror, trying to flash the photographer with the sunlight. How would you be holding the mirror? Do any of the waves seem to you like they’re at that extreme angle?