A string stretched between two rigid supports A and B is set into oscillation by sending a progressive wave from A towards B. The incident wave has amplitude A0, angular frequency ω, and wave number k.
(a) Treating the reflection at B as perfectly rigid and at A as the point of generation, derive an analytical expression for the stationary wave that is ultimately formed on the string.
(b) Show that the distance between successive nodes is kπ.
(c) Prove that the time-averaged power transmitted through any cross-section of the string vanishes once the stationary pattern is established.