A flexible string of length L is fixed at both ends and is made to vibrate at one of its natural frequencies by an oscillator attached near one end. The oscillator supplies waves of constant amplitude A and angular frequency ω that travel along the string and reflect at the boundaries.
(a) Provide a qualitative explanation for why certain points on the string (nodes) remain stationary while others (antinodes) experience maximum oscillation amplitude.
(b) Derive an expression for the wavelength λn of the nth standing‐wave mode (n=1,2,3,…). Clearly state any assumptions used in your derivation.
(c) The tension in the string is doubled while its length and mass per unit length remain unchanged. Determine the factor by which the frequency of the third harmonic changes, and justify your reasoning.
(d) For the fundamental mode, sketch a graph of instantaneous displacement versus position along the string at three instants separated by one‐quarter of the oscillation period. Use the sketches to discuss how energy is exchanged between kinetic and potential forms within one full cycle.