Obtain the equation of frequency of stationary wave propuced in open pipe and show that all harmonics are possible in it.

Explanation

In open pipe antiodes are produced at both ends and consecutive antinodes are obtained at
`(lamda)/(2), lamda, (3lamda)/(2), …, (n lamda)/(2)` where,` n = 1,2,3,…`
If wavelength is `lamda` for open pipe of length L, then stationary waves will only be produced if `L = ( n lamda )/(2)`
`therefore lamda _(n) = (2L)/(n)`
Frequency of stationlary waves in open pipe,
`(v)/( v _(n )) = (2L)/(n) [ because v = lamda v implies lamda = (v)/(v ) ]`
`therefore v _(n) = (nv)/(2L)...(1)` where v is wave speed.
First 4 harmonics are shown for stationary waves in open pipe in figure,

If wp place `n =1 `in equatin (1), then fundamental or first harmonic is obtained,
`therefore v _(2) = (v)/(2L)`
Second harmonic is obtained by putting `n =2` in equation (1),
`therefore v _(2) = (2v )/( 2L) = v/L OR 2 ((v )/(2L)) = 2v _(1)`
Accordingly by putting `n = 3,4 ,...,n` in equation (1), third, fourth, `...n ^(th)` harmonic can be obtained and corresponding (n-1) overtones are obtained.