The meaning of the Kelly-Lochbaum acoustic-tube model

The scattering equations of the Kelly-Lochbaum segmented tube, including the time-varying extension by Strube, are originally based on the assumption of uniform spatial segments and stepwise time update of the acoustic impedances. Here, it is shown that the same equations can be derived without these assumptions for a nonuniform time-varying tube from the discretization of space and time derivatives by the bilinear z transform or by centered differences along the rotated coordinates ct+/-x. Moreover, the same equations also hold for a chain of lattice circuits (or equivalents) with appropriate parameters, if time derivatives are discretized by the bilinear z transform. These circuits can also be extended to simulate uniform segments of varying length.