Brass Instrument Design and the Theory of Horns

Abstract

Engineering the design of horns has been chiefly directed toward transducer applications, where freedom from resonances and a low cutoff frequency are desirable. In the musical horns, easy oscillation and good tuning both call for a high cutoff frequency and accurately placed resonances. The radiation load must not be more than a perturbation at playing frequencies. Horns of rapid flare are required, for which plane-wave theory is inadequate. The “reduced” pressure wave function ψ(z) (=pS12) is mathematically and conceptually convenient, where S(z) is the area of the spherical wavefront at the point z along the horn axis. The “reduced” spherical wave equation ψ″+[k2−U(z)]ψ = 0 (studied by Jansson) is tractable, the horn function U being easily visualized in terms of the transverse and longitudinal curvatures of the horn. Exact solution of the equation is not always required, since useful estimates of U can be made by inspection, whence calculations related to phase integrals, plus perturbation theory permit excellent approximations to the resonance frequencies. Different perturbation methods are useful along the horn depending on local flare. Preexisting horns may be diagnosed and corrected on the basis of laboratory data or “musicians' experiments.” Practical examples will be discussed.