A mandolin string's first harmonic is 392 Hz. How long is the string if the wave speed on it is 329 m/s?

Velocity is given by:

velocity=wavelength*frequency

We note that the wavelength of a wave is twice the length of the string (2w) in the first harmonic.

velocity=329 m/s

wavelength=2w

frequency=392

substituting the values in our equation we get:

329=2w (392)

329=784w

solving for w we get

w=329/784

w = 0.41964~0.42 m