An open organ pipe has fundamental frequency \(100 \mathrm{~Hz}\). What frequency will be produced if its one end is closed? (A) \(100,200,300, \ldots\) (B) \(50,150,250 \ldots .\) (C) \(50,100,200,300 \ldots \ldots\) (D) \(50,100,150,200 \ldots \ldots\)

An open organ pipe produces a harmonic series that includes all integers (1st harmonic, 2nd harmonic, 3rd harmonic, etc.) These harmonics are produced because the air column can support displacement antinodes at both open ends.

Since the fundamental frequency of the open organ pipe is given by \(100 Hz\), we can calculate the frequencies as: 1st harmonic: \(100 Hz\) 2nd harmonic: \(2 \times 100 Hz = 200 Hz\) 3rd harmonic: \(3 \times 100 Hz = 300 Hz\) 4th harmonic: \(4 \times 100 Hz = 400 Hz\) The open organ pipe produces frequencies as multiples of the fundamental frequency: \(100, 200, 300, 400, \ldots\)

A closed organ pipe only supports odd harmonics because the closed end cannot support a displacement antinode. The harmonic series of a closed organ pipe consists only of odd integers (1st harmonic, 3rd harmonic, 5th harmonic, etc.)

When the pipe is closed, the fundamental frequency remains unchanged, \(100 Hz\). However, the pipe will only support odd harmonics now. Since the given frequency is the 1st harmonic (fundamental frequency), we can calculate the 3rd, 5th, and other odd harmonics by multiplying the fundamental frequency by the odd integer: 1st harmonic: \(100 Hz\) 3rd harmonic: \(3 \times 100 Hz = 300 Hz\) 5th harmonic: \(5 \times 100 Hz = 500 Hz\) 7th harmonic: \(7 \times 100 Hz = 700 Hz\) The closed organ pipe produces frequencies as multiples of the fundamental frequency by odd integers: \(100, 300, 500, 700, \ldots\) As per the given answer choices, none of them match the actual answer. However, if the exercise intends to ask for the frequencies produced when an open organ pipe of \(50 Hz\) fundamental frequency is closed on one end, then the answer option (D) would be correct. In that case: 1st harmonic: \(50 Hz\) 3rd harmonic: \(3 \times 50 Hz = 150 Hz\) 5th harmonic: \(5 \times 50 Hz = 250 Hz\) 7th harmonic: \(7 \times 50 Hz = 350 Hz\) The closed organ pipe produces frequencies as multiples of the fundamental frequency by odd integers: \(50, 150, 250, 350, \ldots\) which matches with answer option (D).