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?

• Have you seen a demonstration that for small displacement a taught string obeys the wave equation? – DanielSank Mar 19 at 2:30
• Consider a finite length of string. For each infinitesimal mass element, you write down its equation of motion (i.e $F=ma$). You would obtain a partial differential equation (PDE) known as wave equation. Solve the PDE with your boundary conditions and initial conditions. Then you get what you want. – K_inverse Mar 19 at 2:33
• It is easy enough to show that standing waves form with a discrete spectrum. It's a bit harder to show that the rest of the spectrum is strongly damped. – dmckee Mar 19 at 2:41
• What do you mean by "from first principles"? There might be several derivations to reach this conclusion. – nicoguaro May 23 at 16:06
• Why should first principles be unique? – user7778287 May 24 at 21:53