How does this movement change of the guitar steing is plucked harder?
Answers
The speed of any mechanical wave is generally determined by two things: how hard it is to get the medium moving into order to establish the wave's oscillation (i.e. the mass of a section of the medium), and how strongly the medium snaps back when moved (i.e. the string tension or rigidity of the material). These are both properties of the material and do not depend on how the wave is created. A higher mass (per part) medium has parts that are harder to get moving into a wave pattern, and therefore has a slower wave speed, all else being equal. A more rigid material has parts that snap back quicker when displaced, and therefore has a faster wave speed, all else being equal. The speed of any mechanical wave is a combination of these two effects. The fastest waves therefore travel in low-mass, high-rigidity materials (high-tension in the case of strings).
Plucking a string harder does not change the mass or tension of the string, and therefore does not change the wave speed. It just changes the amplitude of the wave on the string (which leads to the "loudness" of the sound created by the string). This is because plucking just starts the wave. It does not dictate how the wave continues down the string. This is determined only by the properties of the string.