Using your video and the pic with the yellow zebra, it seems that the vertical wave at the beginning is going down at about 1 m/s: taking an estimate of the slowing down of the video at 2.5, and a wavelength of about 1m. For this transverse wave, the tension would thus be 0.001 N, in other words peanuts, if I consider a line weighting approximately 1 gram per meter.
Then you make a soft transition to a casting motion, but in between the nature of your motion has changed, even if visually this does not seem to be so. Let’s review the tension history of a cast from a scientific publication (Gatti-Bono – Perkins).
There are three parts: the acceleration of the line up to maximum speed, the shaping of the loop with the quick compression making difficult to shape a loop with a soft running line, and since the fly leg keeps on moving forward the disappearance of slack is felt through a tug in the line, and finally the rollover of the loop at a more or less constant speed for the major part of it.
- Tension history fly cast.JPG (42.11 KiB) Viewed 2701 times
If you want to consider that all the sequence of line acceleration before loop shaping is (part of) a transverse wave which would embrace the whole cast and line flight, then the corresponding tension values indicate the vertical speed of loop propagation at any time. For 1.2 N, that would correspond to SQRT (1.2/0.001) = SQRT (1200) = 34.6 m/s for the transverse wave: either it goes down to the ground immediately, either it rockets towards the sky at this point in time. This is nonsense obviously and tells us that we do not face a wave here, but the throwing of a projectile to take your words. Such level of tension can be calculated from rod deflection or line mass and its acceleration.For a transverse wave to propagate, two conditions must be met. There must be tension in the medium and the medium must have mass. With those, we can calculate the velocity of propagation of the wave as V = sqrt(Tension/Mass per unit length).
The line tension is increasing during the throwing phase and any disturbance like a dip in the tip trajectory will start propagating in the fly leg: does the vertical transverse wave explain the propagation of tailing loops (horizontal transverse wave) in the fly leg?
Same after loop shaping, any wobble from the tip will follow the loop on the rod leg as long as the cast is tethered. Wobbles also are transverse waves travelling horizontally. The loop is the wave, propagating approximately horizontally; it is a hybrid type of wave, something like 60% longitudinal and 40% transverse, following the definition of such waves (displacement relative to the medium).
You can also consider mends performed with the rod tip, they send transverse waves in various plans according to the motion of the caster whilst the loop is propagating.
I totally disagree here; you can explain the propagation of an untethered line without gravity at all, just using Newton’s laws (momentum, energy). I can’t see how a transverse vertical wave can explain that but would be please to learn, for me only the dynamics of the (horizontal) loop can do it. If you take gravity on board, it will contribute to loop morphing which is not really easy to calculate and this is often left aside given the complexity.Gravity provides the tension vector required for a transverse wave to propagate in an untethered fly line.
When I speak of overall complexity of your explanation, I refer to the fact that there is no need to use the vertical transverse wave concept: you throw a line, and it rolls over thanks to a loop (the only wave we would like to see during the cast). That is simple to understand, one can use a rope lying straight and his arm to watch it: translate the rope forward with increasing speed and stop it to create a loop. I cannot imagine explaining that we should start by wig wagging the rope until we can make a transition to a cast. With a fly rod I can do the same with circles and move progressively to a Belgium cast, and finally to an overhead cast when the large circle will be flat and thin enough. For sure, I am not going to explain the virtues of the circle to comment the overhead cast.
Merlin
When a whip is cracked that is a mighty improvement of dat dere rotation stuff Dude. 
