Oh now this IS an interesting subject and one that will require abit more research!
The Royal Wave has certainly changed over the years and these days it is less affected and more like a normal wave.
I am sure somewhere there must be an etiquette guide showing the do's and don'ts of waving properly.
Back in the olden days, royals used to do a kind of "bicep curl" with the palm facing upwards and gently raising the arm at the elbow. Several Japanese Imperial Royal family members seem to hold their hand very close to the chest with the palm facing outwards and simply moving it back and forth in a very demure way. Here is Britain, the hand seems to be held quite high, palm facing outwards with just the minimal of movement.
Queen Margrethe of Denmark often raised both hands aloft and waves vigourously to the crowds on her birthday balcony appearance.......
I think the Royal Wave technique is different, depending on the circumstances. For instance, simply getting out of a car and walking towards the entrance of the place they are visiting requires a quick, cursory wave of acknowledgment. Standing on a balcony I think requires something a little more dramatic so that the wave can actually be seen far below.
Personally, when I am greeting visitors or sending people off, I tend to stand at the front door with my hand held high and wave enthusiastically from the wrist with minimal arm movements. I went on a cruise ship and when it set sail I was rather more enthusiastic than usual and waved madly at the people standing on the quayside (a small group of men realing in the rope, a couple of custom officers and the remains of a brass-band quartet), but even so, it's nice to wave! Oddly, I also wave to my cat to try and get it's attention (usually ignored) and one always waves at babies in prams!