12) A shower stall measures 88.0cm X 88.0 cm X 215 cm. If you were singing in th

Question

12) A shower stall measures 88.0cm X 88.0 cm X 215 cm. If you were singing in the shower, whichfrequencies will sound the richest (because of resonance). Assumethat the stall acts as a pipe closed at both ends, with nodes atopposite sides, and that the voices of various singers range from130 Hz to 2000 Hz. Let the speed of sound in the hot shower stallbe 355 m/s.
Hz (lowest frequency across shorterdistance)
Hz (second lowest frequency across shorterdistance)
Hz (highest frequency across shorterdistance)
Hz (lowest frequency across longerdistance)
Hz (second lowest frequency across longerdistance)
Hz (highest frequency across longerdistance)

Explanation / Answer

This resonance problem requires the understanding of standingwaves, which happen when a wave is constrained at one or two pointsand the rest of the wave is forced to fit to these points. Ina length of pipe closed at both ends, these standing waves musthave anti-nodes (max/min values) at the ends. These wavesdescribe pressure, so the anti-nodes are like fronts of high andlow pressure bouncing off the ends. Thus the simplest case is one in which a half- wavelength fits inthe pipe with anti-nodes at each end, and a single node (alwayszero) in the middle. The equation that describes this is 1 = 2*L Where 1 is known as the "fundamental wavelength."The fundamental frequency is then, f1 = v / 1 = v/(2*L) Where v is the speed of sound (355 m/s). You can't fit any wavelengths in the tube longer than1 and get resonance, but you can fit1/2, 1/3, 1/4,etc. and still fit the requirement for resonance (standing wave,anti-nodes at the end). Halving the wavelength doubles the frequency so we get fn = n*v/(2*L) This gives nth resonant frequency of the pipe. Also known as (n-1) "octaves" up from the fundamentalfrequency. Now apply this equation using each dimension (longer and shorter)for L, since you are modeling each set of walls as a closedpipe. You can stick any value of n in the above equation and it willstill be a resonant frequency, but you will be limited by the givenrange of the singers' voices.

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