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Exact and approximate analytical time-domain Green's functions for space-fractional wave equations.
Luke M Wiseman1, James F Kelly2, Robert J McGough1
1Department of Electrical and Computer Engineering, Michigan State University, East Lansing, Michigan 48824, USA.
The Journal of the Acoustical Society of America
|September 2, 2019
Summary
This study derives exact and approximate time-domain Green's functions for space-fractional wave equations, revealing stable probability distributions. These functions accurately model ultrasound wave propagation in biological tissues.
Area of Science:
- Physics
- Applied Mathematics
- Ultrasound Imaging
Background:
- The Chen-Holm and Treeby-Cox wave equations model power law attenuation in wave propagation.
- These space-fractional partial differential equations are causal but exhibit differing phase velocities.
- Phase velocity differences impact the shape of time-domain Green's functions, crucial for wave analysis.
Purpose of the Study:
- To derive exact and approximate closed-form time-domain Green's functions for space-fractional wave equations.
- To analyze the behavior of these Green's functions using ultrasound parameters for breast and liver tissues.
- To compare analytical approximations with numerical calculations for accuracy and efficiency.
Main Methods:
- Derivation of exact and approximate closed-form time-domain Green's functions.
- Inclusion of symmetric and maximally skewed stable probability distribution functions in the expressions.
- Numerical evaluation using ultrasound parameters for breast and liver, with reference calculations via the Pantis method.
Main Results:
- Derived Green's functions incorporate both outbound and inbound propagating wave components.
- The inbound component of the wave is found to be negligible at short distances from the origin.
- Single-term analytical expressions involving stable probability densities offer excellent approximations.
Conclusions:
- The derived analytical Green's functions accurately represent wave propagation described by the Chen-Holm and Treeby-Cox equations.
- Stable probability density functions provide a powerful tool for approximating complex wave phenomena.
- These findings have implications for accurate modeling and simulation in ultrasound-based imaging and diagnostics.
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