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Consider a cylindrical pipe open at both ends. We define the acoustic pressure as the difference between the air pressure and the equilibrium atmospheric pressure. To a first approximation the acoustic pressure is zero at the open ends of the tube. The acoustic pressure inside the air column, however, may be non-zero and change with time. The air column can therefore contain standing waves giving a sinusoidal pressure amplitude along the length of the tube such that an integer number of half wavelengths fit into the tube length.

This condition is obviously fairly crude since it ignores the radiation of sound from the ends of the tube. However, in reality, the acoustic pressure amplitude is generally much higher in the tube than outside, meaning that this condition gives a reasonable first approximation for the wavelengths of sound which lead to resonance. These standing waves have frequencies which are integer multiples corresponding to the set of musical pitches we call the harmonic series. The harmonic series consists of a fundamental, the pitch an octave above the fundamental, the octave and a fifth (or twelfth) above the fundamental, the double octave above the fundamental and so on.

Now going on to consider a cylindrical pipe open at one end and closed at the other, our first approximation indicates that the pressure is only required to be zero at the one open end. The air column can therefore contain standing waves such that an odd number of quarter wavelengths fit into the tube length. The fundamental has a frequency half that of a tube of the same length open at both ends and the frequencies are the odd components of the harmonic series.

This thesis has moved to Jonathan Kemp Thesis at http://www.kempacoustics.com/thesis

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