What is your conception of the sine wave in TKD and katas in karate?

The sine wave is primarily supposed to demonstrate settling that should occur in the stance as the technique is applied, which is usually a hand technique of a strike, lock, or throw. Therefore, it's not a relaxation technique. It could, in theory be used as a relaxation technique, but, many of our forms have deliberate slowly-executed sequences, and these should be used to take the opportunity to not only relax, but also a "slow-motion-instant-replay" of how the sequence is executed, with the obvious demonstration of how the rest of the body should move.

As to katas, I can only speak for Taekwondo, but they are organized in terms of increasing complexity - be it internal or external (WTF's Ilyeo, for instance, is nearly all breathing techniques).

As to their purpose... well, forms by nature are training exercises. So their purpose depends on the instructor's objective at the moment: it can be as simple as for warm ups, or practice for balance, stepping, and/or plyometric execution. Of course, they are also libraries of techniques. I also occasionally teach forms the way a psychologist might use Rorschach (or ink-blot) tests: just put it out (or a sequence subset of the form) and let the students analyze it and come up with their own interpretations.

Since I distinguish between "fighting" and "self-defense" as distinctly different concepts, and since I also describe forms as a "self-defense" training aid, I therefore wouldn't classify them as patterns for fighting or sparring. An exception would be if an instructor invented his/her own form for the purpose of sparring training.

And as to whether they are designed for single- or dual- (or more) person opponents, that, again, depends on the interpretation of techniques - and there can be many interpretations for each one. So a given sequence may assume a single attacker, or, could assume multiple - it depends on the interpretation of the sequences at that moment in time. such details might alter your eye line, your speed, and interpretation of successive movements, while still maintaining the integrity of the form/kata.

I'm not familiar with your term sine wave. But kata is not to be practice solely solo as many people believe. You practice it solo and it becomes part of your. But you must also break down the kata and learn the applications of it. This is done with 2 people. You must drill this as well. This is as important as learning the kata. In fact you don't really know the kata if you do not do this. This is vital in kata training. Kokoro often says kata without bunkai is pointless. I don't think I;ve quoted him, but you should be able to get the gist of what is meant.

How can you learn how to execute a throw or pressure point, or joint lock if you don't use drills? These drills should be based upon application based on your kata.

The sine wave as far as I know and have seen is a tkd concept and not a larate consept

There are two man kata, an example is naihanchi, ten no kata, and several others in karate. There are several jojitsu and kenjitsu styles that have two man kata as well.

Kata is the manual to your style every thing you need is in kata, including all your techniques and tactics as well as footwork, etc . Without kata you have no karate.

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when i say two man kata i do NOT mean ONE person doing kata, there are kata that require two individual not all kata is solo. there is an omote an an ura side to several kata. this is not including the bunkai, and other levels of application.

kata is the foundation of many style.

there is far more to kata then your limited interpretation.

Katas are a vital part of training. They force you to make the moves and combinations within them natural to you. So, in an emergency, you have the right defense and offense at hand, without having to think of what is appropriate. They also build stamina and are good mental, as well as physical exercise.

Alright I am A5th DAn in Goju -rRyu ,and 3rd Dan in Judo I know some Krav Maga from the Green Beret training.Listen I'm 51 but look and act30 something.Kata's prearranged form of fighting is necessary to build your stamina and your dexterity.Gotta do them ,listen to your Sensi

It is a way to use gravity to add force to a technique and in some ways it can be used to add speed to a technique much like the sign wave in a whip..

I find it is often the most mis-applied, misunderstood techniques in Taekwando.