A church organ is a musical instrument that creates sound by forcing air through pipes, causing them to resonate. Different notes are created by using pipes of different length. The intensity of a sound wave is proportional to both the square of the amplitude and the square of the frequency. The perceived volume of a sound is indicated by the sound level, measured in decibels: β = 10 log(I/I 0 ), where I is the intensity of the sound, and I 0 is a reference intensity, taken to be 10 –12 W/m 2 .

The characteristic of the sound of a note are determined by the relative intensity of the harmonic frequencies. For pipes open at one end and closed at the other end, the harmonics are determined by the formula nλ = 4L, where n = 1, 3, 5, ……. . For pipes open at both ends, the harmonics are determined by the formula: nλ = 2L, where n = 1, 2, 3, ……. . In both cases, λ is the wavelength of the harmonic, and L is the length of the pipe. The fundamental frequency determines the notes and corresponds to the wavelength where n = 1.

Organs can be tuned by adjusting the length of the pipes. Tuning is generally done with the aid of a pitch pipe, a small pipe than can be blown to produce notes in a standard scale. There notes are then compared to those produced by the organ pipes.

(i) Consider an organ pipe closed at one end. What is the ratio of the frequency of the n = 3 harmonic to the fundamental frequency?