General solution for a horizontal string represents a wave travelling to the right summed with a wave travelling to the left. Case of a string with one end fixed to a wall. A wave travelling towards the wall is reflected antisymmetrically as a wave travelling away from the wall. A wave travelling in a string of finite length produces periodic motion. Yes, write motion as a product of a spatial function and a function of time.
General solution is sum of sinusoids.
Application of boundary conditions leads to transcendental characteristic equation. Eigenfunctions are sinusoids. Solution for time part of synchronous motion corresponding to each eigenvalue. Solutions are necessarily harmonic. General solution to string motion problem is the infinite sum of the synchronous motions combined with undetermined constants.