Formulation of the Brass-Instrument Bore Problem

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Abstract
The boundary conditions for waves in a brass instrument require a pressure amplitude large in the mouthpiece and small at the open end. The nth vibrational mode of the air column therefore consists of (2n−1) standing quarter-wave segments For a horn whose cross section is given by πr2(x), Webster's equation is [∂2/∂x2−U(x)]ψn=−(ωn/c)2ψn. Here, ψn=πr2pn(x), and pn(x) is the pressure distribution for the nth mode, whose oscillation frequency is ωn. U(x)=(d2r/dx2)/r is the product of the longitudinal and transverse curvatures of the horn at x. The similarity with Schrödinger's equation gives considerable insight. WKB methods show a useful relation between the length Ln of a cylindrical pipe (whose nth frequency matches that of the horn) and the value of x at the point where U(x)=(ωn/c)2. For music, the resonances should approximate a harmonic series, so Ln=L1⋅(2n−1)/n. The choice of useful horn types is discussed, including effects of bell shape on starting and maintaining self-sustained oscillations. Effects of bore discontinuities and estimates of radiation and wall damping are described. Experimental data are presented and musical implications drawn.

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