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I feel really dumb for asking this question. I apologize in advance.

When the example of a string attached to a wall is given to explain waves and the ressonance frequencies, the length of the string is given by $L$, which is the length of a straight line in the $x$ axis from the hand to the wall. How can that be if it is an arc? Shouldn't it be higher?

Example in a Walter Lewin lecture.

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If the string is fixed between two points (like two walls), we always assume some elastic material for exactly this reason. When you pluck the string, it becomes longer due to the stretching, which produces a restoring force, that causes it to vibrate.

Very analogous to a pendulum, when you pull on the string, you introduce potential energy into the system, stored in the stretching of the string. Upon release, it contracts back into its original state, but at that point all the energy has been converted to kinetic energy of the string, so it "overshoots" its resting position and deforms to the other side. Just like a pendulum.

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