Determination of Phase Jumps in the Measurement of Phase Velocity of Samples Obeying a Frequency Power-Law Attenuation Coefficient Using Kramers–Kronig Relations

Ultrasonic phase velocity spectroscopy is a very sensitive technique used in the measurement of material properties. In a phase velocity calculation, ambiguities can arise in the spectral phases, in the form of integer multiples of 2π rad, which, if not corrected, results in large errors. In this work, we propose a method for determining these ambiguities, more specifically, the number of 2π rad phase jumps, using the Kramers-Kronig relations, for samples exhibiting a frequency power-law attenuation coefficient. The method is based on a first estimate of the phase velocity from group velocity and attenuation coefficient that are not affected by phase jumps. This estimated phase velocity is used to obtain the number of 2π rad phase jumps, which in turn is used to calculate the corrected phase velocity. The method was tested with samples of liquids with a frequency power-law attenuation coefficient (exponent yvarying from 1.5 to 2) and a solid [polymethyl methacrylate (PMMA)] with y ~ 1, and velocity dispersions ranging from 0 to 34 (cm/s)/MHz.