A stretched string of initial length L is fixed at both ends and vibrates in its third harmonic.
(i) State the relation connecting the wavelength of the standing wave with the length of the string for the nth harmonic.
(ii) The string is now clamped at a point so that only one-half of the original length remains free to vibrate, while the tension and linear mass density remain unchanged. Explain qualitatively how the wavelength of the new standing wave compares with the original wavelength.
(iii) With the help of suitable reasoning, describe how the number and spacing of nodes and antinodes along the vibrating segment change as a consequence of halving the length.
(iv) Comment on whether the frequency of the vibration increases, decreases, or remains the same and justify your answer.