Resonance/Harmonic frequencies in open/closed pipes

[Doc Z]

Homework Statement

The fundamental frequency of an open organ pipe corresponds to E above middle C (329.6 Hz on the chromatic musical scale). The third resonance of a different closed organ pipe has the same frequency. What are the lengths of the two pipes?

Homework Equations

natural frequency of an air column open at both ends - f=nv/2L

and

natural frequency of an air column closed at one end - f=nv/4L

The Attempt at a Solution

I figured out the length of the open pipe by using the first equation. My answer is .520m.

Now, for the closed pipe, since the third resonance frequency of the closed pipe is the same, I used the second equation in the following way - f=3v/4L. However, this is wrong because I should not be using the third harmonic. The question references the 3rd resonance, not the third harmonic, but I have no idea where to go from here. I don't even know what equation to use. Any suggestions?

Thanks for the help!

 

[Doc Z]

Wow I sure feel smart...the third resonance=fifth harmonic! lol sorry guys.

 

[baba89]

I would like to clarify that the term "resonance frequency" can be used interchangeably with "natural frequency" in this context. Both refer to the frequency at which a system naturally vibrates or oscillates. In this case, the fundamental frequency of the open pipe and the third resonance frequency of the closed pipe both have a frequency of 329.6 Hz.

To find the length of the closed pipe, we can use the equation f=nv/4L, where n is the resonance number (in this case, n=3), v is the speed of sound, and L is the length of the pipe. Rearranging the equation, we get L=3v/4f. Plugging in the values for v (the speed of sound in air is approximately 343 m/s) and f (329.6 Hz), we get L=2.61 m.

Therefore, the lengths of the two pipes are 0.520 m and 2.61 m for the open and closed pipes, respectively.