An open pipe is suddenly closed at one end with the result...

An open pipe is suddenly closed at one end with the result that the frequency of third harmonic of the closed pipe is found to be higher by $100 \mathrm{~Hz}$ than the fundamental frequency of the open pipe. What is the fundamental frequency of the open pipe? (A) 240 (B) 200 (C) 300 (D) 480

Key Concepts

Relationship Between Frequency, Speed of Sound, and Pipe Length

Both types of pipes rely on the relationship between the speed of sound, the pipe length, and the wavelength of the standing waves formed in the pipe. This relationship determines the harmonic frequencies, as the wavelength is directly linked to the physical dimensions of the system and the boundary conditions imposed.

Boundary Conditions in Acoustic Resonators

The boundary conditions at the ends of the pipe (open or closed) define where pressure nodes and antinodes occur. These conditions critically affect which harmonics can form and their respective frequencies, thereby influencing the overall resonance behavior of the acoustic system.

Harmonics in Open Pipes

In an open pipe, both ends support antinodes for pressure variations, allowing the formation of harmonic frequencies at integer multiples of the fundamental frequency. The fundamental frequency is given by v/2L, where v is the speed of sound and L is the length of the pipe, and the overtones occur at 2v/2L, 3v/2L, etc.

Harmonics in Closed Pipes

In a closed pipe (open at one end and closed at the other), the closed end supports a node and the open end an antinode. As a result, only odd harmonics are present. The fundamental frequency is given by v/4L, with the successive harmonics occurring at 3v/4L, 5v/4L, and so on.

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