A closed organ pipe of length L is vibrating in its first...

A closed organ pipe of length $L$ is vibrating in its first overtone. There is a point $Q$ inside the pipe at a distance $7 L / 9$ from the open end. The ratio of pressure amplitude at $Q$ to the maximum pressure amplitude in the pipe is
(A) $1: 2$
(B) $2: 1$
(C) $1: 1$
(D) $2: 3$

Key Concepts

Standing Waves

Standing waves are patterns of vibration that result from the interference of two waves traveling in opposite directions. In pipes, standing waves are formed when the reflected sound waves meet the incoming waves, establishing regions of maximum and minimum amplitude, known as antinodes and nodes respectively.

Boundary Conditions in Pipe Acoustics

In an organ pipe, the type of boundary conditions at each end (open or closed) determines the position of nodes and antinodes. Typically, a closed end acts as a displacement node (and a pressure antinode) because the air cannot move, while an open end behaves as a displacement antinode (and a pressure node) due to the free movement of air.

Harmonics and Overtones in Closed Pipes

Closed pipes support only odd harmonics, meaning their resonant frequencies occur at odd multiples of the fundamental frequency. The first overtone in such a pipe is actually the third harmonic. The harmonic mode shapes are determined by the boundary conditions and the harmonic number, affecting the spatial distribution of pressure and displacement.

Pressure Distribution in Standing Waves

The pressure variation along a vibrating pipe in a given mode is described by a sinusoidal function that satisfies the imposed boundary conditions. The location of pressure nodes (points of zero pressure variation) and antinodes (points of maximum pressure variation) are directly related to the harmonic being played. This spatial pattern is critical for understanding how the pressure amplitude varies with position in the pipe.

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