For homebrew multilayer compact coils, I'm not convinced that wave-winding confers any advantages over scramble winding, except that if several coils are wave-wound, with the same number of turns on the same type of former, their inductance will be identical or very close. In terms of inductance, from my limited experience, I've found that wave winding as opposed to scramble winding using the same number of turns, using the same gauge and types of wire on the same former, produces a coil of lower inductance.
Back in 2012, I built a hand wave-winder and wound several test coils. If you look at a commercially produced wave-wound coil, you'll note that the turns are laid close-wound, with each successive turn adjacent to the next one. However, with a home-brew winder, the turns tend to be more spaced. I've tried several adjustments o the angles of the waves for a given width of coil, and different shaped cams - it makes no difference. At the pics below, there's a test coil I wound - not with any particular inductance in mind. (It turned out to be 3 mH). As will be seen, there are gaps between the turns in the waves. The first pic shows the start of winding, with 40 turns would on and the coil taking shape. The second pic shows the finished coil, prior to dipping it in beeswax (which, incidentally, did not alter the inductance).
As to calculating how many turns are called for when winding a coil on a chosen coil former to attain a desired inductance, personally, I wouldn't bother doing any sums - I'd just wind say 100 turns on, and check the inductance. If it's too low, add turns, if it's too high, remove turns. It's all well and good theorising, but experience in life teaches us that 'the difference between theory and practice in practice', is greater than 'the difference between theory and practice in theory'. Fine, perhaps for a single layer close-wound coil of a given diameter, given number of turns of a given wire gauge, but as soon as we get into multilayer coils, life gets more complicated, especially if tapped, where mutual inductance comes into play, so a tapping point at say 10% of turns will not be 10% of total inductance.
I built a 'coil coverage test unit' from a G.A. French 'Suggested Circuit' in Radio Constructor to enable me to check the coverage of homebrew coils in parallel tuned circuits in combination with a variable capacitor. At the time, I was using my homebrew 'Gingery' wave-winder to try to replicate the Repanco DRR2 ('Dual Range Reaction' 2 Band) coil to build the 1959 BBC 'Focus' crystal/2 transistor radio - a very poorly written BBC article aimed at youngsters. If any ever got built, they would have been a big disappointment compared say to a one-valver of that era with reaction. (Basically, a crystal set with a two transistor amplifier tagged on, with very poor selectivity - half a dozen stations, all on top of each other). Pics 3 & 4 show my version of the 'Focus' radio, for which I made a PCB rather than use the messy layout of the 1959 BBC article.
These links might be of interest:
Professor Coyle calculator:
http://www.crystalradio.net/professo...coylecyl.shtml
Homebrew Coil Coverage Test Unit:
https://www.vintage-radio.net/forum/...d.php?t=114238
Homebrew ‘Gingery’ Universal hand wave winding machine YouTube video:
https://www.vintage-radio.net/forum/...d.php?t=106723
Note at the end of the video, how the turns of the finished coil - neat though they are - are not close-wound.
Post 7 at this link – Repanco DRR2 Coil:
https://www.vintage-radio.net/forum/...=REP+DRR2+Coil
Hope that's of interest.