Closed organ pipes Look at the picture of a clarin e t , shown in firgure . It is a pipe with one closed and the other end open . If one end of a pipe is closed , the wave reflected at this closed end is `180^@ ` out of phase with the incoming wave . thus there is no displacement of the particles at the closed end . THerefore , nodes are formed at the closed end and antinodes are formed at open end .
(a) No motion of particles which leads to nodes at closed end and antinodes at open and ( fundamental mode ) ( N- node , A - antinode )
Let us consider the simplest mode of vibration of the air column called the fundamental mode . Anti - node is formed at the open and and node at closed end . from the figure .Let L be the length of the tube and the wavelength of the wave produced . for the fundamental mode of vibration ,we have
` L = ( lamda_1 )/( 4 ) or lamda_1 = 4L`
The frequency of the note emitted is
`f_1 = ( v)/( lamda_1) = ( v)/( 4 L)`
which is called the fundamental note :
Teh freqencies higher fundamental frequency can be produced by blowing air strongly at open end . Such frequencies are called overtones .
THe figure (b) shows the second mode of vibration having two nodes and two antinodes ,
` 4L = 3 lamda _2`
`L=(3lamda _2)/(4 ) or lamda _2 = ( 4L )/(3) `
The frequency for this ,
` f_2 =( V) /( lamda _2 ) = ( 3v )/( 4 L) = 3 F_1`
is called first over , since here , the frequency is three times the fundamental frequency it is called third harmonic .
The figure (c ) shows third mode of vibration having three nodes and three anti - nodes
We have ` 4L = 5 lamda_3`
` L= (5lamda_3) /( 4 ) or lamda_3 = ( 4 L )/( 5 )`
THe frequency `f_3 = ( V)/( lamda _3 ) = ( 5v )/( 4 L) = 5 F_1`
is called second over tone and since n=5 here , this is called fifth harmonic , hence the closed orgran pipe has only odd harmonics and frequency of the nth harmonic is ` f_n = ( 2 n +1) f_1` : Therefore the frequencies of harmonic are in the ratio
` F_1 : f_2 : F_3 : F_4 :... = 1:3:5:7:.............(3)`
