https://www.boatsafe.com/calculate-distance-horizon/
The higher your height of eye (HoE), the farther you can see to the "horizon". This does include radar as it is line-of-sight technology.
When a sub is at periscope depth your periscope HoE is 4-6ft, iirc. That's with the keel somewhere near 60ft iirc (for US boats). Now bring the keel up to "surface-running" at around 15ft draft and you boosted the HoE of your periscope by 45ft.
Its been my experience while playing SH4 that when I/the skipper, while having the periscope up, have my attention turned elsewhere, someone else in the crew is using the periscope to report contacts. I can be looking at the tactical plot and all of the targets are showing up as ships instead of lines of bearing from sonar regardless of their bearing in relation to the viewing angle of the periscope.
The testing should prove that the lookouts can see farther than the skipper can while using binocs or UZO UNLESS the sub being used for the test has had the sail/conning tower cut down putting the lookouts at the same HoE as the skipper.
It's really all just simple math!
