Terhardt, E. (1987). Evaluation of linear-system responses by Laplace-transformation - Critical review and revision of method. Acustica 64, 61-72
Todate utilization of Laplace-transformation for evaluating linear-system responses, particularly for solving pertinent linear differential equations, is intimately dependent on the so-called initial values included in the derivation theorem. Those initial values ordinarily are interpreted as descriptors of the system's initial state. However, conceptual and mathematical analysis of the derivation theorem has revealed that the initial values, if they formally occur, either are null or invalid. Therefore they cannot be used to account for a system's particular initial state, and the pertinent methods todate offered in the literature get questionable in that respect. This is not quite surprising, as application of those methods has already revealed certain inconsistencies anyhow. In the present study the origins of those inconsistencies are analyzed, the foundations of utilizing Laplace-transformation are reviewed, and the consistent method is developed and justified. The new method takes advantage of the fact that the derivation theorem does not include any initial values, and provides separate solutions to the transmission- and the initial-state problems. The method is consistent with the conventional mathematical theory of differential equations, and is formally identical to that of Fourier-transformation.