Understanding Velocity in Waves

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Homework Statement


I was told that a string on a guitar vibrates at a frequency of 110 Hz. The length of the string is .648 m and the tension is 100 N. I was told to find the mass per unit length, μ, of the string. Although I got the correct answer, I was wondering why my initial attempt was incorrect (shown below).

Homework Equations


[itex]f_{n}[/itex] = [itex]\frac{n}{2L}v[/itex]

[itex]v[/itex] = [itex]\sqrt{\frac{T}{μ}}[/itex]

The Attempt at a Solution


Handwritten work in two columns: left column derives f_n = (n/2L) sqrt(T/mu), then f_1(2L) squared = T/mu, mu = 100 / ((110)(2)(.648))^2 = 0.00492 kg/m in green; right column derives v = sqrt(T/mu), 345 = sqrt(100/mu), mu = 100/345^2 = 0.00084 kg/m in red.


(Sorry for the formatting, I don't type in latex really quickly).

In this case, if it's a guitar, why is the velocity of the wave not equivalent to 345 m/s (343.2 m/s)? Why is it so much lower (142.56 m/s)?

Thank you.
 
Oh, is the velocity referring to the velocity of the vibrating string?

I think I understand now.