Numerical evaluations of time-domain Green's functions for the Treeby-Cox space-fractional wave equation with the Pantis method

The attenuation of ultrasound propagation in soft tissue follows a frequency-dependent power law. To model the power law attenuation while also accounting for the dispersion in soft tissue, Treeby and Cox proposed a space-fractional wave equation. Calculating the time-domain Green's function of the Treeby-Cox wave equation requires numerical evaluations of an inverse Fourier transform in the spatial domain, where a highly oscillatory function is integrated from 0 to infinity. This problem is solved by partitioning the integral into two parts with the Pantis method. Time domain Green's functions in breast (y = 1.5, α₀ = 0.086 Np/cm/MHz^1.5, and c₀ = 1450 m/s) at two distances, r = 1 nm and r = 10 cm, are computed to demonstrate the effectiveness of this approach.