Many ITF-based taekwondo practitioners come to notice a difference between different ITF-based organizations with some groups teaching a direct, forceful style and others teaching a more timed, flowing style often referred to as a "sine wave" style. What are the defining characteristics of this "sine wave" approach to ITF hyungs/tuls and how did this style originate?
By reading the The Encyclopedia of Taekwon-do written by General Choi Hong Hi, the founder of Taekwon-do, one would find out that the sine-wave is a main characteristic of Taekwon-do and is part of the "Theory of power" that characterizes the style.
The sine-wave emphasizes relaxation on the upward motion and the contraction and explosion on the downward(final) motion. This allows you to put the weight of your body into the technique.
You can read more about the theory of power here: http://www.saskgtf.com/theory.html
Also the differences you see between ITF sine-waves in different practitioners is because the sine-wave has evolved over time and not all teachers will teach it the same way on each move.
The "sine wave" terminology is used, in my experience, to refer to a more bouncing method of footwork in forms. A linear, non-size-wave execution of kata, such as this performance from Shotokan, values direct movement, crisp end positions, and minimal up-and-down movement between stances. In contrast, this "sine-wavey" performance from ITF TKD exemplifies the up-and-down, more fluid method that the term refers to. Notice the exaggerated way that the practitioner settles into each step, and the alternating slow-fast cadence of each movement. This is dissimilar from the rather even pace that the Shotokan stylist uses.
I am not familiar with the origin of this style, except to note that preferences regarding the look and feel of kata performance naturally change over time and with different personalities in charge. But the salient differences seem to be: