Organ pipes Acoustic waves in tubes encounter different boundary conditions based on whether the tubes are open or closed on the end. If the end is closed, then we know the total velocity at that point must be zero u(Xend)-0. (e.g it is a velocity node) Contrariwise, if the end is open, then we can approximate the excess pressure at that 0-and since pressure is spatially out of phase with position (and velocity), that implies that the total velocity is maximal there (e.g. it is a velocity point as zero P end antinode (a) Compute the reflection coefficients for closed and open ends. For the closed end, approximate the impedance în the second medium as inћп1ty and at the Open end approximate it as zero. Explain how your results are consistent with the node/antinode argument above. (b) For a pipe of length L, derive an expression for the allowed w of acoustic waves in the pipe for the three tube-end cases of open-open, closed-closed, and open-closed. Comment on the similarity/diffcrence in your answers in the different cascs. (c) A tube in the open-closed configuration supports the note middle-C (f- 261.6 Hz) when excited in air (c 346 m/s). The tube is the shortest tube that supports that note. What is that length? ing frequency f be? lower is the resulting note in helium (d) If the tube was instcad cxcited in helium c- 972 m/s), what would the correspond- (e) One octave is a frequency change by a factor of 2. How many octaves higher or (f) Discuss the connection to what it sounds like were you to fill your lungs with helium and try to sing/spcak?