A pipe 20cm long is closed at one end. Which harmonic mode of the pipe is resonantly excited by a 430 Hz source ? Will the same source be in resonance with the pipe if both ends are open ? ( speed of sound in air is `340ms^(-1)`)

Explanation

Here, `l= 20 cm=0.2m , v_(n) = 430 Hz,v= 340 m//s `
The frequency of `n^(th) ` normal mode of vibration of closed pipe s
`v_(n) =( 2n-1) (v)/( 4l)`
`:. 430 = ( 2n-1) ( 340)/( 4 xx 0.2)`
`2n-1= ( 430 xx 4xx 0.2)/( 340)= 1.02 `
`2n =2.02 , n = 1.01 `
Hence it will be the`1^(st)` normal mode of vibration . In a pipe, open at both ends we have
`v_(n) =n xx( v)/( 2l) = ( n xx 340 )/( 2 xx 0.2) = 430`
` :. n = ( 430 xx 2 xx 0.2)/( 340 ) =0.5 `
As n has to be an integer, therefore open organ pipe cannot be in resonance with the source.