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A shower stall has dimensions 86.0 cm xx 86.0 cm xx 210 cm. If you singing in this shower , which frequencies would sound the richest ( because of resonance) ? Assume the stall acts as a pipe closed at both ends , with nodes at opposite sides . Assume the voices of various singers range from130 Hz "to" 2000 Hz. Let the speed of sound in the hot air be 355 m//s. |
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Answer» Solution :We except one set of equally spaced resonance frequencies for sound moving horizontally , and another set with narrower spacing for sound moving vertically . Standing waves must fit between the sides walls , with nodes at both surfaces , or else between floor and ceiling. For a closed box , the resonant air vibrations will have nodes at both sides , so the permitted wavelengths will be defined by ` L = nd_(N N) = (n lambda)/(2) = ( n v)/( 2 f), with n = 1, 2, 3 ,....` Rearranging , and substituting `L = 0.860 m `, the side to side resonant frequencies are `f_(n) = n (v)/( 2L) = ( 355 m//s)/( 2 ( 0.860 m))` ` = n ( 206 Hz)`, for each `n from 1 "to" 9` . With `L' = 2.10 m` , the top to BOTTOM resonance frequencies are `f_(n) = n ( 355 m//s)/(2( 2.10 m)) = n ( 84.5 Hz)` for each `n` from ` 2 "to" 23` We found the allowed values for `n` by checking which resonant frequencies lie in the VOCAL range between `130 Hz "to" 2000 Hz`. If the equations in this problem look like formulas for a pipe open at bothends or a string fixed at both ends , you should draw pictures of the vibrations instead of just thinking about formulas . The shower stall is different from any organ pipe , because it has nodes at both walls instead of antinodes . The shower stall is different from a string from a string because the air wave is LONGITUDINAL . An equation like `d_(N N) = lambda //2` is true for all one - dimensional standing waves , so the best ` f = v//lambda`. A function generator connected to a loudspeaker , with you ear as detector , is all you need to do a nice experiment to observe this PHENOMENON in any SMALL room with hard parallel walls . Listen at different distances from one wall , and you may even discover whether you ear responds more to air displacement or to air pressure . |
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