Show that only odd harmonics are present in the closed pipe system.

Explanation

Let 'L' be the length of the closed pipe.A pipe with one end closed is knonw as a closed pipe system. Let 'v' be the velocity of sound in air. Length of half segment `= ( lambda_(0))/(4)`
i.e., `L = ( lambda_(0))/( 4) `i.e., ` lambda_(0) = 4L`
The least mode of vibration is called fundamental made i.e., fundamental frequency `f_(0)= (v)/( lambda) = (v )/(4L)`
In the second mode of vibration

Here, `L = (lambda_(1))/(2) + ( lambda_(1))/( 4) = (3)/( 4) lambda_(1)`
or `lambda_(1) =( 4)/( 3) L`
The I overtone `f_(1) = (v )/(lambda_(1)) = (3v)/( 4L ) `
In the third mode of vibration ,
The II overtone

`f_(2) = (v)/(lambda_(2))` and `lambda_(2) = (2(lambda_(2)))/(2)+( lambda_(2))/( 4) = ( 5 lambda_(2))/( 4)`
i.e, `f_(2)= ( 5v)/( 4L)`

Hence fundamental frequency `:` I overtone II overtone `: :1 1: 3: 5` Harmonics are odd multiplies of the fundamental `( n-1)^(th)` overtone `= (2n + 1) f_(0) =`where ` n = 0,1,2,3,"....."`