Applications of the causality condition to acoustic scattering

Abstract

The causality condition states that the response of a passive system cannot precede the cause. Under certain conditions, the causality condition leads to a Hilbert transform relation between the magnitude and phase of the complex Fourier transform of a system’s response. This relation has profound implications for those attempting to design passive structures whose desired scattering characteristics are expressed in the frequency domain. Unless the causality condition is satisfied in the frequency domain, the structure is not physically realizable. In this presentation, some novel applications of this relation are developed for a one-dimensional fluid-loaded structure which scatters incident sound in the backward and forward directions. In each application the reflection and transmission coefficients, which are the complex Fourier transforms of the reflected and transmitted pressures due to an impulsive incident pressure wave, are subject to the causality condition. The Weiner–Lee transform, which is derived from the Hilbert transform but is more easily implemented numerically, is used to find complex reflection and transmission coefficients given only their frequency-dependent magnitudes. By using this informa- tion and structural reciprocity, one can find an impedance matrix of a structure which scatters sound in a specified way.