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 | 22nd Mar, 2021, 09:20: AM

Expert Answer:

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
 
begin mathsize 14px style lambda space equals space fraction numerator 4 L over denominator open parentheses 2 n plus 1 close parentheses end fraction end style
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

Answered by Thiyagarajan K | 22nd Mar, 2021, 11:15: AM