On the applicability of Kramers-Kronig dispersion relations to guided and surface waves

Abstract

In unbounded media, the acoustic attenuation as function of frequency is related to the frequency-dependent sound velocity (dispersion) via Kramers-Kronig dispersion relations. These relations are fundamentally important for better understanding of the nature of attenuation and dispersion and as a tool in physical acoustics measurements, where they can be used for control purposes. However, physical acoustic measurements are frequently carried out not in unbounded media, but in acoustic waveguides, e.g. inside liquid-filled pipes. Surface acoustic waves are also often used for physical acoustics measurements. In the present work, the applicability of Kramers-Kronig relations to guided and surface waves is investigated using the approach based on the theory of functions of complex variables. It is demonstrated that Kramers-Kronig relations have limited applicability to guided and surface waves. In particular, they are not applicable to waves propagating in waveguides characterised by the possibility of wave energy leakage from the waveguides into the surrounding medium. For waveguides without leakages, e.g. those formed by rigid walls, Kramers-Kronig relations remain valid for both ideal and viscous liquids. In the former case, Kramers-Kronig relations express the exponential decay of non-propagating (evanescent) higher-order acoustic modes below the cut-off frequencies via the dispersion of the same modes above the cut-off frequencies. Examples of numerical calculations of wave dispersion and attenuation using Kramers-Kronig relations, where applicable, are presented for unbounded media and for waveguides formed by rigid walls.