wavelength, frequency, harmonic of its fundamental vibration

Question

An open organ pipe is 0.84 m long. What is the wavelength of its fundamental vibration? What is the frequency of its fundamental vibration. What is the frequency of the first overtone? What is the frequency of the third overtone. What is the frequency of the third harmonic?

Answer 1

For this pipe I will assume it is open at one end and closed at the other. To see how waves fit in pipes consider this gif.

https://www.acs.psu.edu/drussell/demos/standingwaves/standing.gif

What is the wavelength of its fundamental vibration? 

The fundamental vibration has the longest wavelength. 1/4 of the wave length will fit in the pipe. This will have a node at the closed end for particle displacement and an antinode at the open end of the pipe.

L = 0.84m

L = 1/4 λ

λ = 4 (0.84 m) = 3.36 m

What is the frequency of its fundamental vibration.

v = λ f, where v = 343 m/s (the speed of sound).

f = (343 m/s )/(3.36 m) = 102.1 Hz

What is the frequency of the first overtone? 

What is the frequency of the third overtone.

These would be the next waves that can fit in the pipe.

L = 1/4 λ, 3/4 λ, 5/4 λ, ... (2n-1)/4 λ

λ = 4 L/ (2n-1)

v = λ f

f = v (2n-1)/(4 L)

n = 2, f = 306.3 Hz

n = 4, f = 714.6 Hz

 What is the frequency of the third harmonic?

The third harmonic is the same as the second overtone.

n = 3, f = 510.4 Hz