Department of Medical Physics and Bioengineering, University College London, Gower Street, London, WC1E 6BT, United Kingdom.
J Acoust Soc Am. 2011 Jun;129(6):3652-60. doi: 10.1121/1.3583537.
An efficient Green's function solution for acoustic initial value problems in homogeneous media with power law absorption is derived. The solution is based on the homogeneous wave equation for lossless media with two additional terms. These terms are dependent on the fractional Laplacian and separately account for power law absorption and dispersion. Given initial conditions for the pressure and its temporal derivative, the solution allows the pressure field for any time t>0 to be calculated in a single step using the Fourier transform and an exact k-space time propagator. For regularly spaced Cartesian grids, the former can be computed efficiently using the fast Fourier transform. Because no time stepping is required, the solution facilitates the efficient computation of the pressure field in one, two, or three dimensions without stability constraints. Several computational aspects of the solution are discussed, including the effect of using a truncated Fourier series to represent discrete initial conditions, the use of smoothing, and the properties of the encapsulated absorption and dispersion.
本文推导了一种在具有幂律吸收的均匀介质中求解声学初值问题的高效格林函数解。该解基于无损耗介质的齐次波动方程，并增加了两个附加项。这两个项依赖于分数拉普拉斯算子，分别用于描述幂律吸收和频散。对于压力及其时间导数的初始条件，该解允许使用傅里叶变换和精确的 k 空间时间传播器在单个步骤中计算任意时间 t>0 的压力场。对于规则间隔的笛卡尔网格，可以使用快速傅里叶变换有效地计算前者。由于不需要时间步长，因此该解无需稳定性约束即可在一维、二维或三维中高效计算压力场。讨论了该解的几个计算方面，包括使用截断傅里叶级数表示离散初始条件的影响、平滑的使用以及封装的吸收和频散的特性。
