Physics: frequency, harmonic number

Question

A pop bottle style instrument is played being open at one end closed at the other, if two successive frequencies, 350 Hz and 490 Hz, are produced, what is the fundamental frequency, and what harmonic number are these two frequencies listed.

Answer 1

f_n = frequency of nth harmonic (n = 1,3,5,7,... for open/closed pipe)

 

f_n = v / lambda_n

 

L = length = n (lambda_n / 4) for the nth harmonic

 

lambda_n = 4 L / n

 

f_n = v / (4 L / n) = nv / (4L)

 

Harmonics of open/closed pipe are odd numbers only, so consecutive harmonics are separated by 2.

f_n+2 / f_n = [(n+2) v / (4L)] / [nv / (4L)] = (n+2) / n = 490 Hz / 350 Hz = 1.4

 

(n+2) / n = 1.4

 

n = 5

 

Therefore, the 350 Hz frequency is the 5th harmonic and the 490 Hz frequency is the 7th harmonic.

 

f_5 = 5 f_1 so f_1 = f_5 / 5 = 350 Hz / 5 =70 Hz

 

Similarly, f_1 = f_7 / 7 = 490 Hz / 7 = 70 Hz

 

The fundamental frequency is f_1 = 70 Hz.