Applications of the causality condition to one-dimensional acoustic reflection problems

The causality condition is examined as a means of determining frequency-domain information about a submerged object from a partial knowledge of its acoustic reflection characteristics. A one-dimensional problem is considered in which an acoustic wave reflects from an object that is described by the impedance it presents to the fluid. Two new applications of the causality condition to the frequency-domain analysis of this problem are investigated and illustrated by numerical examples. In each application, the causality condition is used to find the object's complex impedance from a knowledge of the reflected wave's magnitude. The first application is to experimental studies where one desires a knowledge of an object's complex impedance but practical limitations only allow a measurement of the reflected wave amplitude. Analysis shows that the causality condition may be used to determine the phase of the reflected wave, and hence the object's impedance, if the reflection coefficient is minimum phase. When this is true, examples suggest that the phase of the reflection coefficient may be accurately determined from the causality condition even in the presence of noise and band-limited data. The second application is to design situations, where one wishes to create an object that reflects sound with a specified frequency-dependent magnitude. The causality condition may aid the designer by providing a knowledge of all causal object impedances that produce the same reflection coefficient magnitude. A numerical example is presented in which a variety of causal object impedances produce the same reflection coefficient magnitude over an infinite frequency range.