To find out mode of vibration of air in a resonating pipe

[opencircuit]

Homework Statement

A cyclindrical pipe of length 29.5 cm closed at one end resonates with a tuning fork of frequency 864 Hz. Then mode of vibration of air in the pipe is(Velocity of sound in air=340m/s):

The attempt at a solution

V=Velocity, n=Frequency

If l₁ is the first resonating length
then l₁ = V/4n = 340/(4X864) = 0.0983m = 9.83 cm

Also, l₂ is second resonating length
l₂ = 3V/4n = 3 X 9.83 = 29.51 cm

This is the maximum resonating length possible in the pipe. Therefore the mode of vibration is 2.
But the answer given is 3! Please help...

 

[bacon]

Given a pipe with one open end and one closed, can you have n=2? and, more generally, can you have an even n?

 

[lightgrav]

how many quarter-waves are in this pipe?

 

[opencircuit]

Given a pipe with one open end and one closed, can you have n=2? and, more generally, can you have an even n?

No the frequency(n) is not 2. The mode of vibration is 2. So there's a difference.

how many quarter-waves are in this pipe?

l₂ = 3λ/4
So, there are 3 quarter waves, right?

 

[bacon]

The mode of vibration is the "harmonic". The second allowed harmonic is not necessarily the second harmonic. In this case, the second allowed harmonic is the "third harmonic" corresponding to n= 3. If n=2 were allowed, the second harmonic, there would be a node at the open end of the pipe, which can't exist.
I hope this makes sense.

 

[opencircuit]

The mode of vibration is the "harmonic". The second allowed harmonic is not necessarily the second harmonic. In this case, the second allowed harmonic is the "third harmonic" corresponding to n= 3. If n=2 were allowed, the second harmonic, there would be a node at the open end of the pipe, which can't exist.
I hope this makes sense.

Ok I get it. Pipe closed at one end has only odd harmonics. And the second resonance actually occurs at the third harmonic(i.e. the third mode of vibration).Thanks

 

[opencircuit]

Hey second resonance does occur at the third harmonic but it is still the second mode of vibration( harmonic is not necessarily same as the mode of vibration)

I got this reference in a book:
Second normal mode of vibration : n2 = v/λ2 = 3v/4L = 3n1
(pipe closed at one end)

I think the answer given is wrong...