# How is the phase velocity for a wave in a stringed instrument derived?

I was searching on the internet for a formula relating velocity of wave and force of tension of a string (eg how the tension in a string instrument effects the velocity) and this formula keeps appearing.

$$v_{\text{wave}} = \sqrt{\frac{F_{\text{tension}}}{{\rho}_{\text{linear}}}}$$

What is the derivation of this formula as I can not find it online?

• en.wikipedia.org/wiki/String_vibration#Derivation – Nat May 7 '18 at 2:42
• Probably, "P" is a misreading of Greek rho $\rho$ for linear density. – user137289 May 7 '18 at 2:45
• Just to note it, the above Wikipedia link answers this question, but it's still neat to have such information added to StackExchange. That said, I'm far too impatient to transcribe a derivation at the moment, so anyone who feels like it, please feel free! (It's also 100% acceptable for askers to self-answer, if you care to review the derivation by explaining it to others, @Ditto. The $\mathrm{\TeX}$ can largely be obtained by copy/pasting Wikipedia's $\mathrm{\TeX}$ expressions and wrapping them in \$'s.) – Nat May 7 '18 at 2:58