Sunsets at the Beach and Streetlights in Windshields

Some time ago, Hannah asked me if I knew the explanation of the effect one sometimes sees while driving at night, where streetlights look like they have rays of light emanating from them in two opposite directions, which rotate as you drive by. I didn't know, but, after a conversation, we decided that it's very related to the phenomenon that causes the reflection of the sun in the ocean at the beach to look like a path of light heading out to the horizon rather than a small circle of light like one would expect from a mirror.

The thought is that the ocean is choppy rather than flat, and so the surface of the water is often tilted away from the horizontal, up to some maximal angle that depends on how big the waves are. The sun is way brighter that anything else that the ocean might reflect, so, even if only a small portion of the light hitting your eye from a particular point came from the sun, it will drown out any other stuff reflected from there.

If the ocean were flat, there would be some point in the water where light from the sun would bounce off and hit your eye, so you'd see the reflection of the sun there. However, if you move that point in the water either towards the sun or towards you, it doesn't take a very big change in the angle of the water to compensate for the change to ensure that light from the sun still hits you after bouncing off. Since the ocean is choppy, we can expect there to be a little piece of water surface somewhere around there at the right angle to do that for you. It helps that the sun is a reasonably sized disk, so there's some room for error. So, we can expect to see a bright path in the water heading from you towards the sun.

OK, but why don't we see a reflection of the sun everywhere else, too? The problem is that, as you move the point in the water to the side, the angle of the water needed to bounce that light back to you quickly becomes quite large. When that angle becomes larger than the maximal angle of the waves, you stop seeing a reflection of the sun. So, you'd expect the path of the light to be fairly narrow, though it would get wider with a choppier ocean, and also it would get a bit wider near the point of "true" reflection.

So, what's this have to do with windshields? Well, light entering glass gets bent according to Snell's Law. When the two sides of the glass are flat and parallel, like in a windshield, once the light leaves the glass, it gets bent back to the same direction it was originally going. (A lens, on the other hand, exploits having the two sides of the glass be at angles to each other in order to make a net change in the direction of the light.) However, nothing is perfectly smooth and flat, and so some light will hit an imperfection in the surface of the glass where the surface is at a slightly different angle.

Like with the ocean, we can assume that there's a maximum angle of the imperfections in the glass. The details of the double-refraction are different, but the result is the same as with the ocean: after going through the glass, the light will be off by at most a small angle from where it is "supposed to" go. In fact, if we turn the ocean into glass and turn the sun into a streetlight that is below the surface rather than above it, we should see the same "path of light" effect that we see at the beach, though with a thinner path since the glass is less "choppy" than the water.

The only problem left is that the ocean is horizontal, whereas (unless you're very bad at driving) a car's windshield is not, and it's a bit tricky to do 3D rotations in your head to convert one to the other. Fortunately, there's a convenient trick for figuring out which direction to expect the line of light to point. At the beach, the path of light follows the line joining your feet to the point in the water where you'd expect to see the "true" reflection of the sun. You can think of your feet as a magical solar light path attractor: The path of light from the sun will point towards your feet wherever you go.

With a car windshield, we can do the same thing. Instead of your feet, you need to imagine a line from your eye that is perpendicular to the windshield. Since windshields are tilted, this line will go upwards and forwards from your eye and probably hit the plane of the windshield above the actual windshield itself. That point is your magical streetlight line of light attractor: You can expect that the line of light from the streetlight will stay pointed towards your magical line of light attractor as you drive by.

Now, being a theorist, I haven't actually gone out and verified that the direction of the line of light is as I described. I'd be curious to hear if folks see it.