# 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?

Apr 30, 2018

Given,

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

Three harmonics, all odd multiples of $25$.

${f}_{9} = 225 \text{Hz}$, ${f}_{11} = 275 \text{Hz}$, 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 ${f}_{1} = 25 \text{Hz}$, and the pipe must be,

f_1 = nu/(4ℓ)

=> ℓ = nu/(4f_1) approx 3.43"m" long.