Show that both odd and even harmonics are present in an open pipe system.

Explanation

A pipe with both ends open is called open pipe system.
I mode of vibration `:`
Let L be the length of the open pipe system.
Let v be the velocity of sound inair .
and `f_(0)` be the fundamental frequency .

from the fig `L =( lambda_(0))/( 2) `i.e.,`lambda_(0) = 2L`
But `f _(0)= ( v)/( lambda_(0)) = ( v )/( 2L )` .......(1)
II mode of vibration `:`
Here `L = 2 ((lambda_(1))/( 2)) =lambda_(1)`
Hence I overtone `f_(1) = ( v)/( lambda_(1)) = ( 2v)/(2 lambda_(1)) = 2f_(0)`

III mode of vibration `:`
Here `L= 3 ( ( lambda_(2))/( 2))`
so that II overtone `f_(2) = ( v )/(lambda_(2))`
or `f_(2) =( 3V )/( 2L)`

Hence, the fundamental , I overtone , II overtone are in the ratio `f_(0) : f_(1) : f_(2) : : 1: 2: 3`.
The `n^(th) `harmonic `= ( n -1)^(th ) ` overtone = `n f_(0)` .