title: The acoustical Klein-Gordon equation: the wave-mechanical step and barrier potential functions.
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.