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show that in the case of closed organ pipe, the ratio of the frequencies of the harmonics is 1:3:5:7.​
Asked by ks076345 | 22 Mar, 2021, 09:20: AM
Modes of vibrations to create sound wave in a tube of one closed end and other end open are shown in above figure

Wavelength λ is related to length of tube as follows

where n = 0, 1, 2, ...... etc

If v is velocity of sound wave and f is frequency , then we have

frequency f = v / λ  = (2n+1) × [ v / (4L) ]

If n = 0 , fundamental mode frequency  fo = [ v / (4L) ]

If n =1, first harmonic frequency f1 = 3 × [ v / (4L) ]

if n =2 , second harmonic frequency f2  = 5 × [ v / (4L) ]

so on

Hence ratio of frequencies of harmonics = 1 : 3 : 5 : 7 ....... etc
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