Time domain analysis of causal and noncausal fractional wave equations

The attenuation of ultrasound propagating in human tissue follows a power law with respect to frequency. Several different models for power law attenuation have been developed, where many of these are partial differential equations that contain fractional derivatives in time or space. Some of these models are causal, and some are noncausal, yet all of these models describe attenuation that follows a power law in the frequency domain. To demonstrate the similarities and differences in the time domain responses predicted by the causal and noncausal models, Green´s functions are calculated numerically for the power law wave equation and for the Caputo fractional wave equation. These Green´s functions are evaluated numerically for liver with a power law exponent of y=1.139 and breast with a power law exponent of y=1.5. Simulation results show that, although the power law attenuation observed in the frequency domain is comparable over the range of ultrasound frequencies of interest, the time domain responses for these fractional wave equations differ in the nearfield region. Furthermore, the noncausal features of the numerically calculated time domain response, if present, are only evident in the extreme nearfield region. Simulation results also show that, the causal and the noncausal Green´s functions converge to the same time domain waveform in the farfield. Thus, in the context of these two time-fractional wave equations, causality is essentially a phenomenon of the extreme nearfield, and the difference between causal and noncausal solutions is insignificant elsewhere.