waves on a plucked string (answers)
1-3. Here is the key.
4. The wave is a standing wave because there are nodes which remain in a fixed location.
5. The animation shows what you should obtain.
6. The slope of either side is constant for the first quarter cycle. At the equilibrium position, the slope switches sign but then remains constant for the next quarter cycle.
7. The slope of the plateau is 0 and does not change with time.
8. The animation is similar for about the first quarter cycle. Just before it reaches the equilibrium position, the sides begin to fall behind the plateau. Past the equilibrium position, the sides flip over. After they do, the slope is no longer constant but rather increases as the string approaches its maximum upward displacement. Moreover, the plateau isn't completely flat anymore.
9. The wave speed is greatest when the string is at its points of maximum displacement. Here is where the tension is greatest and the linear density least.