A violin string of uniform linear density is mounted between two fixed supports. The tension in the string is adjusted so that transverse waves travel along it at v=2.60×102ms−1. The designer wishes the fundamental (first-harmonic) frequency of the open string to match the musical note middle C, which has a frequency of 261.6Hz. After tuning, the violinist shortens the effective vibrating length by pressing the string against the fingerboard 12cm from the upper support.
(a) For the open, untouched string, determine the wavelength associated with the desired fundamental standing wave.
(b) Using your answer to part (a), calculate the total length the string should have between its fixed supports.
(c) When the violinist shortens the vibrating portion by 12cm, determine the new fundamental frequency produced (assume wave speed on the string remains the same). Express the result to an appropriate number of significant figures.
(d) Explain why varying the amplitude of vibration would not, by itself, alter the pitch of the note produced, even though it changes the loudness.