The conical bore in musical acoustics
R. Dean Ayers, Lowell J. Eliason, Daniel Mahgerefteh · American Journal of Physics · 1985
A survey of descriptive textbooks on musical acoustics shows a need for a legitimate, convincing explanation for the seemingly paradoxical behavior of the conical bore. We examine the differential equations and their solutions for spherically symmetric waves in three dimensions, settling on acoustic pressure as the variable of interest because it behaves more simply than particle velocity or displacement. Next we solve the boundary-value problem for all combinations of open and closed ends on truncated cones. The natural frequencies so obtained are then justified with plots of the pressure standing waves; it is at this point that we find a physically correct explanation that can be shared with students in a descriptive acoustics course. This pictorial approach is also used to explain the shapes of the input impedance curves for a straight pipe and a frustum. We conclude with a careful analysis of the successive distortions to which a pressure pulse is subjected as it bounces back and forth inside a frustum.