An organ pipe has a length of 50 cm. Find the frequency of its fundamental note and the next harmonic when it is: 

(a) Open at both ends. 

(b) Closed at one end (speed of sound = 350 m s⁻¹). 

(Ans: (a) 350 Hz, 700 Hz, (b) 175 Hz, 525 Hz)



Data Given:


Length of organ pipe = l = 50 cm = 0.5 m

Speed of sound = v = 350 m s⁻¹


To Find:


(a)  for open pipe
      Fundamental frequency  f₁ = ?  
      2nd harmonics f₂ = ?

(b) for closed pipe
      Fundamental frequency  f₁' = ?  
      3rd harmonics  f₃' = ?


Solution:


(a)  for open pipe
      Fundamental frequency  f₁ = ?  
      2nd harmonics f₂ = ?

The formula for fundamental frequency for open pipe 

f₁ = `\frac {v}{2l}`

putting values

f₁ = `\frac {350 m s⁻¹}{2 ｘ0.5 m}`

f₁ = 350 Hz -----------Ans. (1)

The formula for nth frquency is 

fn = n f₁ 

for n=2

f₂ = 2 ï½˜ 350 Hz 

f₂ = 700 Hz -----------Ans. (2)


(b) for closed pipe
      Fundamental frequency  f₁'  = ?  
      3rd harmonics  f₃' = ?

The formula for fundamental frequency for closed pipe 

f₁'  = `\frac {v}{4l}`

putting values

f₁'  = `\frac {350 m s⁻¹}{4 ｘ0.5 m}`

f₁'  = 175 Hz -----------Ans. (1)

The formula for nth frequency is 

fn = n f₁ 

for n=3 (only odd harmonic) 

f₃'  = 3 ï½˜ 175 Hz 

f₃'  = 525 Hz -----------Ans. (2)



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