In Jarhead and Generation Kill, why do soldiers cover their scopes so there is only a small opening?

It is to keep the glare to a minimum (or eliminate) as to not accidentally give their position away. Also, it is nice to use during the daylight because it acts as an aperture control (for the covers with a small hole) for the lense, since you don't want all the light during the day to come in (such as on a camera lense) and to help protect all of their optics from exposure damage over time to sunlight, especially UV light.

Great shows by the way. If I remember seeing scope covers in Band of Brothers, I assume they would have been complete coverings and the small holes were only seen on the rifle scopes in Generation Kill. Difference obviously in technolgoy with 60 years between the wars.

Cuts down on reflections. If you looked at a hill and saw something shinning (when it should only be trees and brush), you'd be calling in an air strike or lobbing a couple of mortars in the area.

To eliminate the reflection from the glass as much as possible.

Cars have the same thing....a slit cover that coveres the headlight.....well at least the germans had it on their opels and benzes....