Pressure disturbances in air-filled free space travel away from their source at a rate known as the speed of sound. The simplest mode of propagation of sound is in the form of plane waves where the wavefronts are flat two dimensional planes. The pressure variations can then be described entirely by a propagating sinusoidal pressure profile along the third spacial direction, perpendicular to the wavefront planes.
Most musical wind instruments consist of pipes of circular cross-section. In fact many, such as the flute and the clarinet, feature an air column roughly cylindrical in shape. The acoustic waves set up within wind instruments have wavefronts perpendicular to the walls, meaning that the waves inside a cylindrical pipe are plane waves. Instruments whose bore is conical with a small enough apex angle object may also be assumed to contain plane waves in their air column. The air column can then be modelled by approximating the bore by a series of short concentric cylinders whose radius matches that of the instrument at each step along its length. If there are very many very short cylinders the bore of the cylinder series almost exactly matches the bore of the instrument. The plane waves in the cylindrical sections behave like plane waves in free space except that they are partially reflected and partially transmitted by any changes of cross-sectional area within the pipe.
For acoustic horns (such as the brass musical instruments) which feature rapidly flaring air columns, the wavefronts become more spherical in the bell section of the instrument in order to meet the wall at 90 degrees. A calculation of the properties of the instrument may still be made assuming plane wave propagation although the results will be far less accurate than for instruments without any flaring section. The basic theory of sound propagation in a musical wind instruments is set out in the literature mainly assuming plane wave propagation. To go beyond this, the actual pressure field can be expressed in terms of the sum of the plane waves present in the tube and the contribution from the modes of the duct whose pressure profile is not uniform on the plane perpendicular to the central axis. This method is labelled modal decomposition or multimodal propagation.
This chapter comprises of a review of plane wave propagation in pipes of constant cross-section and at a discontinuous change in cross-section. The discussion then moves on to treat multimodal propagation. In order to describe fully the acoustic behaviour of an acoustic horn we must include the acoustic effect of radiation of sound from the open end. Chapter 3 will therefore treat this problem.