A `3 m` long organ pipe open at both ends is driven to third harmonic standing wave. If the amplitude of pressure oscillations is `1` per cent of mean atmospheric pressure `(p_(o) = 10^(5) Nm^(2))`. Find the amplitude of particle displacement and density oscillations. Speed of sound `upsilon = 332 m//s` and density of air `rho = 1.03 kg//m^(3)`.

Answer

(i) `S_(0) = (1)/(1089 pi) m , (ii) (1)/(1089) kg//m^(3)`

Explanation

` 3 = 3(lambda)/(2)`
`lambda = 2 m`
`P_(m) = 100 N//m^(2) , V = 330 m//s , rho_(0) = 1 kg//m^(3)`
i. `P_(m) = Bs_(0) k = rho_(0) v^(2) s_(0) ( 2pi)/( lambda)`
`s_(0) = (lambda P_(m))/( rho_(0) v^(2) 2 pi) = ( 2 xx 100)/( 1 xx 330 xx 330 xx 2 pi)`
ii.`B = -( dp)/( dV//V) = (dp)/( d rho//rho)`
`[ m = rho v rArr 0 = (d rho)/(rho) + ( d v)/( v)]`
`rArr d rho = (rho d p)/(B)`
`( d rho)_(max) = ( rho p_(m))/( rho v ^(2)) = (100)/(108900) kg//m^(3) = (1)/( 1089) kg//m^(3)`