Department of Electrical and Computer Engineering, Michigan State University, East Lansing, Michigan 48824, USA.
Department of Statistics and Probability, Michigan State University, East Lansing, Michigan 48824, USA.
J Acoust Soc Am. 2019 Aug;146(2):1150. doi: 10.1121/1.5119128.
The Chen-Holm and Treeby-Cox wave equations are space-fractional partial differential equations that describe power law attenuation of the form α(ω)≈α|ω|. Both of these space-fractional wave equations are causal, but the phase velocities differ, which impacts the shapes of the time-domain Green's functions. Exact and approximate closed-form time-domain Green's functions are derived for these space-fractional wave equations, and the resulting expressions contain symmetric and maximally skewed stable probability distribution functions. Numerical results are evaluated with ultrasound parameters for breast and liver at different times as a function of space and at different distances as a function of time, where the reference calculations are computed with the Pantis method. The results show that the exact and approximate time-domain Green's functions contain both outbound and inbound propagating terms and that the inbound component is negligible a short distance from the origin. Exact and approximate analytical time-domain Green's functions are also evaluated for the Chen-Holm wave equation with power law exponent y = 1. These comparisons demonstrate that single term analytical expressions containing stable probability densities provide excellent approximations to the time-domain Green's functions for the Chen-Holm and Treeby-Cox wave equations.
Chen-Holm 和 Treeby-Cox 波动方程是空间分数阶偏微分方程，描述幂律衰减形式为 α(ω)≈α|ω|。这两个空间分数阶波动方程都是因果的，但相速度不同，这会影响时域格林函数的形状。针对这些空间分数阶波动方程，推导出了精确和近似的封闭形式时域格林函数，得到的表达式包含对称和最大倾斜稳定概率分布函数。使用乳房和肝脏的超声参数在不同时间和不同距离作为空间和时间的函数进行数值评估，其中参考计算是使用 Pantis 方法计算的。结果表明，精确和近似的时域格林函数包含向外和向内传播项，并且在离原点很短的距离内，向内分量可以忽略不计。还针对幂律指数 y = 1 的 Chen-Holm 波动方程评估了精确和近似的解析时域格林函数。这些比较表明，包含稳定概率密度的单项解析表达式可以很好地近似 Chen-Holm 和 Treeby-Cox 波动方程的时域格林函数。
