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How are stationary waves formed in closed pipes ? Explainthe various modes of vibrations and obtain relations for their frequencies. |
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Answer» Solution :A pipe, Whichis closedat one end and the other is opened is called CLOSED pipe. When a sound wave is SENT through a closed pipe, which gets reflected at the closed end of the pipe.Then incident and reflected waves are in same frequency, travelling in the opposition direction are superimposed stationary waves are formed. To form the stationary wave in closed pipe, whichhas atleast a node at closedend and antinode at open end of the pipe, it is knownas first harmonicin closed pipe. Then lengthof the pipe ( l ) is EQUAL to one fourth of the wave length. `:. l = ( lambda_(1))/(4) implies lambda _(1)=4l` If `'v_(1)'` is fundamental frequency then `v_(1)= ( upsilon)/(lambda_(1))` where `'upsilon'`is velocityof sound in air `v_(1)= ( upsilon)/( 4l)= v `...(1) To form the next harmonicin closed pipe, two nodes and two antinodesshould be formed. So that there is possible to form third harmonicin closed pipe. Since one more node and antinode should be INCLUDED. Then length of the pipeis equal to `(3)/(4)` of the wavelength `:. l = ( 3lambda_(3))/(4)` where `' lambda_(3)'` is wavelengthof third harmonic `lambda_(3) = ( 4l)/(3)` If`' v_(3)'`is thirdharmonic frequency( first overtone ) `:. v _(3)=( upsilon)/( lambda_(3))= ( 3upsilon)/( 4l)` `v_(3) = 3v` ...(2) Similarly the next overtone in the close pipe is only fifth harmonic it will have three nodes and 3 antinodes between the closed end and open end. Then length of the pipe is equal to `(5)/(4)` of wave length `( lambda_(5))` `:. l = ( 5 lambda_(5))/(4)` where `'lambda_(5)'` is wave length of fifth harmonic . `lambdad_(5) = ( 4l)/( 5)` If `'v_(5)'` is frequency of fifth harmonic( second overtone ) `v_(5)= ( upsilon) /( lambda_(5)) =( 5upsilon)/(4l)` `v _(5) = 5v` ...(3) `:. ` The frequencies of higherharmonics can be determined by using the same procedure . Therefore from the Eq (1) , (2) and(3)only odd harmonics are formed. Therefore the ratio of the frequencies of harmonics in closed pipe can be written as `v_(1) : v_(3) : v_(5)= v : 3v : 5v ` `v_(1) : v_(3) : v _(5)= 1: 3 : 5 ` 
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