Prove from first principles that a guitar string will vibrate at a constant frequency From experience I am aware that a taught string will generally vibrate with a constant frequency. I wanted to prove this by considering the relation of distance from the resting position, and its subsequent velocity (or velocities if non-constant) away from that initial extension.  
I realize a suitable end goal would be to prove that a taught string will take the same amount of time to reach its opposite maximum for an ideal range of initial extensions, but I do not have the relevant background to lay out the actual math. What math is involved and how is it composed?
 A: It sounds to me like you're asking about the velocity of the actual string, and not the wave on it.  If I paint a dot on a guitar string and deflect that dot away from equilibrium, it sounds like you're asking whether that dot reaches the opposite maximum displacement in the same interval of time regardless of how far you pull it away from equilibrium.  If that's the question, the answer is yes.  This assumes a linear restoring force on the string, which is a very accurate approximation for actual strings in typical use-case scenarios.  Under such conditions the resulting differential equation has a period that is independent of amplitude of motion.
If you're interested in the formation of a wave on the string, or the rate at which that wave progresses along the string (wave speed), here is a link to a chapter of a book I wrote in which they're derived: https://www.geogebra.org/m/nyssufjx#material/S2NjxeJh.
If you are interested in all the waves that can form on the string, that depends on boundary conditions.  The locations and types of supports affect the answer since you end up with waves bouncing back and forth and meeting up - over and over.  The easiest way to handle that math is with phasor diagrams.
