How to determine resonance of an open or closed pipe?

Homework Statement

A pipe resonates at successive frequencies of 540 Hz, 450 Hz, and 350Hz. Is this an open or a closed pipe?

Homework Equations

L = (nλ)/2 or L = ((2n-1)/4)λ
v = fλ

The Attempt at a Solution

The difference between the first two frequencies (540 & 450) is 90Hz, and the difference between the last two frequencies (450 & 350) is 100Hz.
I have no idea as to how to solve this equation. I tried asking my physics teacher, and we could not come up with an answer, but I am still curious about the solution.
No temperature is given, therefore I cannot determine the speed of sound and rearrange for lambda. Unless I assume that the temperature is room temperature.
I also don't understand how to determine the difference between an open & closed pipe since the equation: L = (nλ)/2 is applicable to both open-open & closed-closed pipes.

Filip Larsen
Gold Member
Welcome to PF!

I agree that the three frequencies and your equations taken together are puzzling. Perhaps there is a typo somewhere or something has slipped out?

Regarding the different relationship between pipe length and resonance frequency for open and closed pipes, you can understand it as a result coming from the different boundary conditions in the two cases. For a closed pipe there can be no air flow across the end thus giving resonance to any sound wave with an integer number of half waves. For an open pipe there can be no (or only very small) pressure difference across the end thus giving resonance to any wave with an integer number of half waves plus a quarter wave. See [1] for some illustrative diagrams of this.

If we look at just the two lowest frequencies then if a pipe resonates at 350 Hz and the next resonance is at 450 Hz you can write this as fn = 350 Hz and fn+1 = 450 Hz, where n is unknown. Using the relationship between pipe length and frequency for either an open or a closed pipe you should now be able to solve and find an expression for n in the two cases. For instance, for an open pipe the pipe length is given as

L = (n/2)(v/fn) = ((n+1)/2)(v/fn+1)
Can you determine whether the pipe is open or closed from the value of n in this case? What condition must n satisfy?

Repeating this exercise for the frequencies 450 Hz and 540 Hz should ideally give same result (open or closed) as for 350 Hz and 450 Hz, but for your numbers it gives the opposite which of course is puzzling.

[1] http://en.wikipedia.org/wiki/Acoustic_resonance