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 third overtone are in resonance with a given tuning fork. The ratio of the length of $\mathrm{P}_{1}$ to that of $\mathrm{P}_{2}$ is: (a) $8 / 3$ (b) $3 / 8$ (c) $1 / 2$ (d) $1 / 3$

Key Concepts

Frequency-Length Relationship

The frequency produced by an organ pipe is inversely proportional to its length. For open pipes, the harmonics occur at multiples of v/(2L), while for closed pipes only odd multiples (v/(4L), 3v/(4L), etc.) occur. This relationship is crucial in determining how the dimensions of the pipe affect the sound it produces.

Overtones

Overtones are the frequencies above the fundamental frequency. The first overtone is the second harmonic, the second overtone is the third harmonic, and so on. The overtone series differs between instruments depending on the boundary conditions, such as whether the tube is open at both ends or closed at one end.

Resonance

Resonance occurs when an external source drives a system at its natural frequency, leading to large amplitude oscillations. In acoustic systems such as organ pipes, resonance ensures that the sound produced is amplified when the frequency of the tuning fork matches one of the pipe’s natural frequencies.

Standing Waves in Pipes

Standing waves in pipes are formed due to the interference of incident and reflected sound waves within the pipe. The boundary conditions—whether the end of the pipe is open or closed—determine the positions of nodes and antinodes, which in turn establish the allowed modes of vibration within the pipe.

Harmonics

Harmonics are the frequencies at which a system naturally vibrates, typically appearing as integer multiples of a fundamental frequency. In musical instruments, these harmonics contribute to the timbre of the sound, as each harmonic reinforces the overall tone when the instrument is played.

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