In a pipe organ, air is forced to flow through hollow pipes with precisely determined lengths. One end of the pipe is always open, to allow air into the pipe. The other end may be opened or closed. (a) If a pipe is to generate a note with a frequency of 256 Hz (middle C) when closed at one end, how long should the pipe be? (b) What are the frequencies of the next two harmonics produced by this pipe? (These are the pitches of the next two standing waves that the pipe will admit.) (c) What fundamental frequency will the pipe produce if both ends are open? (d) What are the frequencies of the next two harmonics when both ends are open? (These are the pitches of the next two standing waves that the pipe will admit.)

When played in a certain manner, the lowest resonant frequency of a certain violin string is 410 Hz. What is the frequency of the (a) second and (b) third harmonic of the string? (a) Number i (b) Number i Units Units

The overall length of a piccolo is 34 cm.The resonating air column vibrates as in a pipe that is open at both ends. In this problem,the frequency of a “note” means the fundamental frequency of the standing waves in theair column. (a) Find the frequency of the lowest note a piccolo can play, assuming the speed of soundin air is 345 m/s.(b) Opening holes in the side effectively shortens the length of the resonant column. If thehighest note a piccolo can sound is 4200 Hz, find the effective length of the resonantcolumn.

You are trying to design a musical instrument that consists of a pipe filled with air at 20oC, so that it produces two successive harmonics that have frequencies of 240 Hz and 280 Hz. The speed of sound in air at 20oC is 343 m/s. (a) Should the pipe be open at both ends, or closed at one end? Justify your response. (b) How long should the pipe be?