An organ pipe P1 closed at one end vibrating in its first...

An organ pipe $\mathrm{P}_{1}$ closed at one end vibrating in its first overtone and another pipe $\mathrm{P}_{2}$ open at both ends vibrating in its third overtone are in resonance with a given tuning fork. The ratio of lengths of $\mathrm{P}_{1}$ and $\mathrm{P}_{2}$ is: (a) $1: 2$ (b) $1: 3$ (c) $3: 8$ (d) $3: 4$

Key Concepts

Resonance

Resonance occurs when two or more systems oscillate at the same natural frequency, allowing them to exchange energy efficiently. In the context of organ pipes, it means that the frequency of the vibrating air column in the pipe matches the frequency of an external source (like a tuning fork), causing the pipe to vibrate in harmony with the source.

Overtones and Harmonics

Overtones are the higher frequency modes of vibration that occur above the fundamental frequency. In musical acoustics, harmonics are integer multiples of the fundamental frequency. The first overtone is the second vibrational mode, but in specific cases—such as a pipe closed at one end—the modes follow a pattern of only odd harmonics, meaning the first overtone actually corresponds to the third harmonic.

Closed and Open Pipe Acoustics

Organ pipes behave differently depending on whether they are open at both ends or closed at one end. For a pipe closed at one end, only odd multiples of the fundamental frequency are produced, while an open pipe supports all harmonics. This fundamental difference leads to different wavelength and frequency relationships based on the physical boundary conditions of each pipe.

Relationship Between Pipe Length and Frequency

The frequency of sound produced in a pipe is inversely proportional to its length. For an open pipe, the fundamental frequency is given by v/(2L), while for a closed pipe it is given by v/(4L). When overtones are used, these relationships are adjusted by the harmonic number. Understanding this relationship is crucial in determining the length ratios when the pipes are in resonance with a given frequency.

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