Question

The fundamental frequency of an open organ pipe is 261.6Hz. The third resonance of a closed organ pipe has the same frequency. If the speed of sound in air is 340m/s, what is the length of the pipes? 

Answer 1

Let's first find the wavelength of the wave from the wave speed equation:

$$v=f\lambda,$$ $$\lambda=\dfrac{v}{f}=\dfrac{340\ \dfrac{m}{s}}{261.6\ Hz}=1.3\ m.$$

a) For the fundamental frequency (or the first harmonic) of an open organ pipe there is the following length-wavelength relationship:

$$\lambda=2L,$$ $$L=\dfrac{\lambda}{2}=\dfrac{1.3\ m}{2}=0.65\ m=65\ cm.$$

b) For the third resonance frequency (or the third harmonic) of the closed organ pipe there is the following length-wavelength relationship:

$$\lambda=\dfrac{4}{3}L,$$ $$L=\dfrac{3}{4}\lambda=\dfrac{3}{4}\cdot1.3\ m=0.975\ m=97.5\ cm.$$