Night before last I was observing at Petit Jean State Park in Central Arkansas. The transparency was so-so, and I bounced around to several brighter Messier objects. I was looking at the Ring nebula with the Stowaway, Baader Mark V, 1.25 GlasPath corrector, and 12.5mm Plossl eyepieces.
With the Ring centered, a satellite (bright dot), went across my FOV from my right to left, and I thought: Nice surprise. THEN, about 10 seconds later, another bright dot went from my left to my right, but its path was about 1/2 way from the center of the FOV to the edge. It moved a little slower than the previous satellite, and it followed a sine curve of about 3-4 cycles. At first I thought it was a firefly, but:
1. It was a steady brightness light-dot
2. It was a dot, not a smudge (it was focused)
3. It was a steady brightness all the way through the FOV (not a flash, etc.)
4. It was the same color as the previous satellite dot
5. It appeared for about 3 seconds duration (longer than a firefly)
6. It appeared to be the same size light point as the previous satellite
Maybe you guys might have an answer. I thought maybe tumbling space junk
or satellite? Or maybe my friend James' alien friends? or? At any rate, it was a first for me to see something like this, and it made a so-so night more exciting.
I have been observing for more than 65 years, off and on. I was sober. I was astonished. The period of the sine wave was constant, with equal peaks and nulls. It was about 10 PM local time. That's about all I can recall.