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Forbes, Barbara J.; Pike, E. Roy and Sharp, David B.
(2003).
DOI: https://doi.org/10.1121/1.1590314
Abstract
The transformed form of the Webster equation is investigated. Usually described as analogous to the Schrödinger equation of quantum mechanics, it is noted that the second-order time dependency defines a Klein–Gordon problem. This "acoustical Klein–Gordon equation" is analyzed with particular reference to the acoustical properties of wave-mechanical potential functions, U(x), that give rise to geometry-dependent dispersions at rapid variations in tract cross section. Such dispersions are not elucidated by other one-dimensional—cylindrical or conical—duct models. Since Sturm–Liouville analysis is not appropriate for inhomogeneous boundary conditions, the exact solution of the Klein–Gordon equation is achieved through a Green's-function methodology referring to the transfer matrix of an arbitrary string of square potential functions, including a square barrier equivalent to a radiation impedance. The general conclusion of the paper is that, in the absence of precise knowledge of initial conditions on the area function, any given potential function will map to a multiplicity of area functions of identical relative resonance characteristics. Since the potential function maps uniquely to the acoustical output, it is suggested that the one-dimensional wave physics is both most accurately and most compactly described within the Klein–Gordon framework.
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