Hello Tater,
I'm still in active naval duty today. I've served some 30 years as a subhunter. Starting as a sonar operator gradually made my way up via helicopter crewmember to chief of the opsroom in an ASW frigate.:p
To spot a feather (wake from a moving subs periscope) is hard when there is a seastate of 4 or more. (about 1 meter waves, little white on top). When the sub does not move in those (or worse conditions) it's virtually impossible to spot a periscope from a ship, unless it's very close. (500 yds or so). In clear weather, a sea like a table, it's obviously a lot easier to spot a periscope, but the sub knows that as well.
On the other hand, spotting a sub from a helicopter has some very interesting points to it. In the Caraibean sea I've seen subs at 40 meter depth when flying at a curtain angle from the sun and at a certain time of day (meaning the sunrays had to penetrate the water in a certain angle.)
Spotting a periscope in rougher weather was just as hard as it is from a ship.
A submarine can be given away by sea life. I've witnessed (more than once I might @) dolphins in a "V" shaped group in the sea. It looked like a living arrow. When we came flying there and stooped to lower our active sonar about 1000yards in front of that "V" shape we promptly had sonarcontact with high doppler. Mind you, the sub was deep, nevertheless the dolphins gave him away.
Flying in that helo we had a searchlight sonar, meaning we could sent soundwaves in a 30 degree angle and not all around, just like the early sonars on board warships in the second WW. This means the sonar operator has to be very alert ALL the time. This is a weak spot on every navy craft, for staying at your tops for a prolonged period is hard.
Same for the aviators. Flying ours and ours over the ocean looking for a sub, a survivor or life raft is extremely hard to maintain and sooner or later your alertness weakens.
So, looking into the water to see a shallow sub is possible when the weather conditions are right. Seeing a periscope in a body of water without waves is possible. At windforce 3/4, seastates 4 or more it's a lucky shot if you ever see one.
Don't know if it helps, just adding to the subject.
Regards,
Bando