A word on string analogy
In the teaching forum Graeme explains to us his interpretation of the mechanisms allowing controlling the line. The basic equation of transverse waves is used: tension in the string equals the linear mass of the line multiplied by the squared speed of the wave in the string. Change one parameter and the other two follow: increase linear mass under given tension and the wave speed diminishes. Increase tension for a given linear mass and the speed increases. Simple and straightforward at first sight, and the reading made me think of Berlin’s explanations (see “a week with Berlin” in the special guest forum).
The same approach is applied to leaders, using impedance as a guideline (impedance is tension divided by speed for a string and equals linear mass multiplied by speed). I recommend to reread wthe discussion on leader tapers.
There is however a big assumption hidden here, because this relationship applies to a string held tight in between two fixed points like a guitar string, there is no free end like for a line, and no rollover as well. At some stage there is a contradiction with known situations: the following one relates to a snap cast:
The linear mass of the string decreases as the wave travels through the taper, so the wave speed increases.
That should be applicable to the end of the rollover then; the tapered line section should show acceleration since the linear mass decrease along the string approach. But if you cast a level line, you can see that the final acceleration is much more pronounced than for a tapered one and ends with a kick back: there is too much energy left in the line. The statement above illustrates the fact that nothing coming from the actual situation of a loop (drag) is considered with the string approach. There is something well known however: as the diameter of a line decreases, drag forces take progressively the lead on line momentum change and at some stage (a small diameter), the line decelerates. You can have both trends following each other as the taper enters the loop: an acceleration trend caused by a favorable change in momentum within the loop, followed by a deceleration due to the size of the line. And this is the one which always wins at the end given the diameter of leaders.
There is a wave speed increase with snap, pull back, and check casts because the caster applies a force on the rod leg which results in an acceleration of the “wave” speed (the tangential velocity around the loop, which I often call the “rotation speed” of the loop). This happens whatever the taper of the line is. The physics behind has little to do with strings, but with the application of Newton laws on the line (loop and legs). The same physics show that the travel speed of the loop is reduced and seems to nearly stop or even go backwards sometimes, but after a short while it reaccelerates again. The taper of the line has its say but it is not the main parameter of this late acceleration.
Among the equations needed to simulate a cast, there is the “angular momentum” (AM) one (recommended by James). Months ago I was on the “string side” but when I realized I was unable to simulate a pull back correctly, I restarted from the beginning and finally found a way to better simulate, and consequently better understand, what is going on with such cast. In this equation, there is a “morphing term”, which is represented by the possible variation of the radius of a semi circular (idealized) loop. This equation is not easy to solve if you do not assume that morphing is negligible, but you do not need to solve it if you want to understand what can happen on the morphing side. An important parameter is the “wave” speed. You can increase it with a pullback and then the AM equation tells you that for a level line the size of the loop should reduce. If on top of that the diameter of the line increases, this trend is reduced and if the diameter of the line decreases, the trend is amplified. As an example, if you pull back late enough during the time the front taper enters the loop, then the loop size should reduce. If you push forward instead, the loop size should increase. There are several possibilities in fact (e.g. PB during the time a reversed taper enters the loop) some of them giving mitigated results (little morphing) in terms of loop size.
Although it is simple and easy to use, the string wave speed equation has been established for a specific situation which is not encountered for a loop (fixed ends). Taking the time (and the pain) to write correctly equations for a variable mass system allows understanding the internal mechanisms governing particular actions like PB of snap. And despite the reasoning with the string approach may conclude sometimes similar things by comparison to the actual physics of the line; it is just inappropriate on the technical ground and can end with false conclusions.
There can be some true transverse waves in the line, like a mend or a dangle, which can travel along the line, reflect at the end like any transverse wave. They belong to a different category by comparison to a loop. To such waves you can apply the string theory, but not to the loop.