Numerical analysis of causal and noncausal power-law Green's functions

Accurate models of attenuation and phase speed are required for simulations of transient linear ultrasound propagation. Ideally, these models should be causal, although several models of ultrasound propagation that incorporate the effects of loss are noncausal. To distinguish between causal and noncausal models, the analytical loss terms that appear in the time domain Green's function for the power law wave equation are analyzed numerically with the STABLE toolbox for three different values of the power law exponent y. The results show that these loss terms for the power law wave equation are causal for 0 <= y < 1 and noncausal for 1 <= y <= 2. Limiting forms of these loss terms are obtained at the source, and in this specific location, the otherwise noncausal Green's functions are causal. However, a short distance from the source, a noncausal response is demonstrated for 1 < y <= 2. Propagation delays are shown to obscure the noncausal behavior of the loss term for y = 1, and causal behavior is consistently observed for 0 <= y < 1.