On the application of the generalized Kramers-Kroenig dispersion relations to ultrasonic propagation

The broad theme of the dissertation is the application of the generalized Kramers-Kronig dispersion relations to ultrasonic propagation. We develop the Kramers-Kronig dispersion relations in the sense of tempered distributions as well as the conventional point function sense. Non-local and nearly-local forms to the Kramers-Kronig dispersion relations are derived for the prediction of dispersion in media with attenuation obeying an arbitrary frequency power law. Furthermore, a time-domain representation of the generalized Kramers-Kronig dispersion relations is developed and compared with a time-causal theory of ultrasonic propagation.;The first aspect of ultrasonic propagation that we investigate is the intimate relation between the ultrasonic attenuation coefficient and phase velocity. Recently there have been concerns expressed regarding the validity of the Kramers-Kronig dispersion relations to media with attenuation obeying a frequency power law. We demonstrate, however, that ultrasonic measurements of systems with attenuation obeying a frequency power are causally consistent. Consequently, valid Kramers-Kronig relations are available. Theoretical predictions for the frequency dependence of attenuation and phase velocity compare favorably to experimental measurements for a series of liquid specimens over a range of temperatures and acoustic pressures.;The second aspect of ultrasonic measurements we investigate is the phenomenon of phase cancellation at the face of a phase-sensitive receiver which is present in measurements of phase-aberrating media. The excess loss due to phase cancellation is well-known, and has long been of interest. What has not been explicitly investigated is the possibility that there exists a phase velocity shift corresponding to this excess loss. In a novel proposal, we relate the excess loss due to phase cancellation to a phase velocity shift via a nearly-local form of the Kramers-Kronig dispersion relations. Furthermore, we provide a measure of the artifact present in a phase velocity measurement using phase-sensitive and phase-insensitive detection techniques. We demonstrate the technique on measurements of textile woven composites using a two-dimensional pseudo-array and a one-dimensional array.