A pipe in air at 20°C is to be designed to produce successive harmonics at 225 Hz, 275 Hz, and 325 Hz. Is the pipe open or closed? How long does the pipe need to be?

Which harmonics are these?

1 Answer
Al E.
Apr 30, 2018

Given,

#nu = (343"m")/"s"# (the speed of sound at #20°"C"#)

Three harmonics, all odd multiples of #25#.

#f_9 = 225"Hz"#, #f_11 = 275"Hz"#, et cetera

For open pipes,

#f_n = n*nu/(2ℓ)#

For half-open pipes (e.g. closed),

#f_n = n * nu/(4ℓ)#

The key to this question is knowing that "closed" pipes must have harmonics in odd multiples, to ensure an antinode being at the open end and a node being at the closed end.

Hence, the pipe is closed , the fundamental frequency is #f_1 = 25"Hz"#, and the pipe must be,

#f_1 = nu/(4ℓ)#

#=> ℓ = nu/(4f_1) approx 3.43"m"# long.

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