Causal theories and data for acoustic attenuation obeying a frequency power law

Abstract

This study compares causal theories, based on Kramers–Kronig relations, fractional calculus, and on those derived from new time domain causal relationships, to diverse data. All these theories are based on the assumptions that the functional form of the attenuation persists beyond the measurement range and that attenuation is much smaller than the wave number. The data, for lossy media with attenuation having a power-law frequency dependence with an exponent y, include cases for both liquids and solids, ranging from acoustic to ultrasound frequencies. Data are in closer correspondence with the new theory which predicts decreasing dispersion as the power exponent y approaches zero or an even integer. Experimental results and supporting evidence show that the classical case of frequency-squared attenuation is dispersionless. An approximate nearly local Kramers–Kronig theory is in agreement with the time causal theory when the exponent is close to one, but deviates for other values. The comprehensive time causal theory is shown to be equivalent to two other theories derived from exact Kramers–Kronig relations and from fractional calculus and it covers the y odd integer cases which are missing or incomplete in these approaches. Attenuation–dispersion relations are presented in two forms: one for a general frequency range and another for a finite range. It is demonstrated that complete velocity dispersion (within a signal bandwidth) can be predicted from knowledge of the attenuation data and velocity at a single frequency including the velocity at either zero frequency (y≳1) and or at a high-frequency limit (0<y<1). In addition, attenuation data can be predicted from velocity data.