Why do overlapping grids make broad moiré bands?

Two window screens can produce broad dark bands even though each screen contains only fine repeated lines. The large pattern seems to belong to neither layer. It appears because their smaller patterns interact: some openings line up, while other openings are partly or completely covered. This changing alignment is called moiré.

Consider two transparent sheets carrying parallel opaque stripes. Where one dark stripe lies over another, the two share much of the same blocked area, leaving more nearby space open. Where the stripes slip out of alignment, one layer blocks openings in the other. Less light can pass through the combined pair. Across the sheets, a gradual change in alignment becomes a broad change in brightness.

A small spacing difference can produce a surprisingly large repeat distance. Suppose one set repeats every 10 drawing units and the other every 11. Their stripe beginnings line up at the starting point. After one interval they differ by 1 unit; after several intervals, their alignment has drifted farther. At 110 units, eleven 10-unit intervals equal ten 11-unit intervals. The complete arrangement can begin again there.

The same result can be expressed with spatial frequency, meaning repeats per unit distance. The two frequencies are 1/10 and 1/11. Their difference is 1/110, so the reciprocal gives a beat spacing of 110 units. The broad pattern is therefore tied to a difference between two close repetition rates. Neither separate sheet needs a 110-unit stripe printed on it.

Sliding one sheet sideways changes where the light and dark bands fall, but it does not change this beat spacing while the two stripe spacings remain fixed. If both sets instead have identical spacing and stay parallel, their relative alignment is the same everywhere. They may cover more or less clear space depending on their offset, but they no longer drift into broad repeating bands across the sheet.

Here the effect comes from one opaque pattern covering part of another. It is not cancellation between light waves. Moiré can also arise with angled lines, dot patterns and sampled images, but their geometry needs its own explanation. The broad lesson from these simple screens is that a tiny mismatch repeated many times can become a large visible pattern—a relationship that makes small changes in spacing much easier to notice.