APPLICATIONS OF THE CAUSALITY CONDITION TO ACOUSTIC REFLECTIONS

摘要

The causality condition, which states that the response ofa passive system cannot preceed its cause, places mathematicalconstraints on the complex Fourier transform of the response. Forminimum-phase-shift systems, these constraints take the form ofHilbert transform relations between the magnitude and phase ofthe Fourier transform. These relations have historically had aprofound impact on the design of electrical circuits, as evidencedin Bode's text Network Analysis and Feedback Amplier Design.The present work presents new applications in the area of structuralacoustics which show how the mechanical impedance of apassive °uid-loaded structure can sometimes be inferred from themagnitude of its acoustic re°ection coecient. The approach isdemonstrated on a one-dimensional problem in which an incidentacoustic wave re°ects from a passive structure. An equivalent formof the Hilbert transform, known as the Wiener-Lee transform, isused for the numerical calculations. If the re°ection coecient isminimum-phase-shift, then its phase can be uniquely computedfrom its magnitude. In this work, it is shown that the minimumphase-shift property of the re°ection coecient actually dependson the real part of the mechanical impedance of the structure.Therefore, in an inversion problem where one attempts to inferthe mechanical impedance given the magnitude of re°ection, additionalassumptions about the structure must be made in order toarrive at a unique answer. In cases where one attempts to design apassive structure with a specied re°ection coecient magnitude,the causality condition yields a mechanical impedance which correspondsto a minimum-phase-shift re°ection coecient.