Research done by thiago Olabarriaga:
1-dimensional wave derivation
The line above on the graph is to represent a string under tension forming a wave. There are the vectors to help guide the derivation. Now, the string does not move on the direction, but moves on the direction, and the string causes tension for both sides equal to each other. Because only moves, the focus may go directly to the direction. An equation may be found the derivate for.
The left side is negative because it is pointing down, and the right side is opposite, so it becomes positive. For small angles, sin the in radians.
Supposed the mass of the string is and to find this mass times it must equal to the force above.