Stationary waves

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Question

Two adjacent natural frequencies of an organ pipe are formed to be 550 Hz and 650 Hz. Calculate the length of this pipe (in m). (Velocity of sound in air 350 m/s).

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Solution

In case of open organ pipe we have all types of harmonic, even as well as odd. While in case of close organ pipe we have only odd harmonic. The ratio of two adjacent frequencies $\frac{{\mathrm{f}}_{\mathrm{n}}}{{\mathrm{f}}_{\mathrm{n}+1}}=\frac{550}{650}=\frac{50×11}{50×13}=\frac{11}{13}$Hence the adjacent harmonic are 11th and 13th. Both harmonic are odd hence it is the case of close organ pipe. Clearly, the fundamental frequency should be 50 Hz.The fundamental frequency in case of close organ pipe${\mathrm{f}}_{0}=\frac{\mathrm{v}}{4\mathrm{l}}⇒50=\frac{350}{4×\mathrm{l}}$Hence length of the pipe $\mathrm{l}=\frac{350}{50×4}=1.75\mathrm{m}$

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