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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?
in Grade 12 Physical Sciences by Diamond (48,764 points) | 23 views

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ν=343ms (the speed of sound at 20°C)

Three harmonics, all odd multiples of 25.

f9=225Hz, f11=275Hz, et cetera

For open pipes,

fn=n⋅ν2ℓ

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

fn=n⋅ν4ℓ

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 f1=25Hz, and the pipe must be,

f1=ν4ℓ

⇒ℓ=ν4f1≈3.43m long.

by Wooden (180 points)

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