Solve problems involving standing wave formation in pipes open at both ends, closed at one end, and on stretched strings using 𝐿 = 𝑛 𝜆 2 and 𝐿 = (2𝑛 − 1) 𝜆 4.

A music student investigates standing waves in two identical pipes, each of length 0.85 m. Pipe A is open at both ends, while Pipe B is closed at one end. Both pipes are made to resonate at their fundamental frequency using a sound source at 200 Hz. The speed of sound in air is 340 m s⁻¹. a) Determine which pipe, if either, will resonate at 200 Hz when driven at its fundamental frequency. b) Justify why the other pipe will not resonate at this frequency by comparing the required pipe length for resonance with the actual pipe length.

A flute and a clarinet are both approximately 0.66 m in length. The flute behaves as a pipe open at both ends, while the clarinet behaves as a pipe closed at one end. Compare the fundamental frequencies produced by these two instruments and explain the physical reason for the difference.

A physics class investigates resonance in air columns using two identical cylindrical tubes, each 0.68 m long. Tube A is open at both ends. Tube B is closed at one end with a rubber stopper. A tuning fork of frequency 500 Hz is held above each tube. The speed of sound in air is 340 m s⁻¹. a) Determine whether the fundamental frequency of Tube A will resonate with the 500 Hz tuning fork. Justify your answer with calculation. b) Predict which harmonic, if any, will resonate in Tube B when the tuning fork is sounded above it.