Causality Difficulties Which can Arise in Modelling the Reflection of a Normally Incident Wave from a Lossy Planar Surface

摘要

Causality difficulties that can arise in modelling the reflection of a normally incident pulse from a lossy planar surface are studied theoretically. These difficulties can arise when the traditional approach to treating acoustic loss mechanisms (i.e., the generalization of lossless formulas via use of complex wavenumbers) is employed instead of using the proper viscoelastic relations. The problem caused by insufficient care being exercised in the choice of frequency dependence of the material properties of the lossy medium. In the present work, three models for the material properties of the lossy medium are considered. In all cases the phase speed in the material is assumed to be independent of frequency. Losses are assumed to be modellable with complex wavenumbers. The loss factor is chosen to be (1) frequency independent, (2) linearly dependent on frequency, and (3) quadratically dependent on frequency. For all three material models, exact analytical expressions for the reflected pressure are obtained. It is demonstrated that the requirements of causality are not satisfied for cases (2) and (3). These results are interpreted mathematically in light of the appropiate Hilbert transform properties required for physically realizable signals. The analytical expressions presented herein for the reflected pressure are useful for investigators who numerically model echo reduction systems.