Analytical time-domain Green’s functions for power-law media

Abstract

Frequency-dependent loss and dispersion are typically modeled with a power-law attenuation coefficient, where the power-law exponent ranges from 0 to 2. To facilitate analytical solution, a fractional partial differential equation is derived that exactly describes power-law attenuation and the Szabo wave equation ͓"Time domain wave-equations for lossy media obeying a frequency power-law," J. Acoust. Soc. Am. 96, 491-500 ͑1994͔͒ is an approximation to this equation. This paper derives analytical time-domain Green's functions in power-law media for exponents in this range. To construct solutions, stable law probability distributions are utilized. For exponents equal to 0, 1 / 3, 1 / 2, 2 / 3, 3 / 2, and 2, the Green's function is expressed in terms of Dirac delta, exponential, Airy, hypergeometric, and Gaussian functions. For exponents strictly less than 1, the Green's functions are expressed as Fox functions and are causal. For exponents greater than or equal than 1, the Green's functions are expressed as Fox and Wright functions and are noncausal.However, numerical computations demonstrate that for observation points only one wavelength from the radiating source, the Green's function is effectively causal for power-law exponents greater than or equal to 1. The analytical time-domain Green's function is numerically verified against the material impulse response function, and the results demonstrate excellent agreement.

Fractal ladder models and power law wave equations

“…49 Note that other FPDE models for power law media, such as the Szabo wave equation 18 and the power law FPDE in Ref. 50, satisfy the same relationship.…”

Time-domain comparisons of power law attenuation in causal and noncausal time-fractional wave equations

“…Similar to the Gaussian function, the stable pdf in the numerator of Eq. (8) is strictly positive 7 for all values of t when y ¼ 1.5, so the power law wave equation is noncausal at all distances for y ¼ 1.5.…”

“…The power-law wave equation, 7 which is closely related to the Szabo wave equation in Eq. (3), is given by…”

“…2 Additional examples of mammalian tissues with various power law exponents are tabulated in the book by Duck, 3 and other compiled lists of attenuation values are found in papers by Goss et al 4,5 The corresponding wave equations that describe power law attenuation in soft tissue utilize fractional derivatives, which are non-integer order derivatives. These fractional derivatives are often time-fractional, [6][7][8] although spacefractional derivatives are also used. 9,10 Examples of timefractional lossy wave equations that model attenuation and dispersion of ultrasound in soft tissue include the Szabo wave equation, 6 the Caputo wave equation, 11 and the power law wave equation.…”

“…9,10 Examples of timefractional lossy wave equations that model attenuation and dispersion of ultrasound in soft tissue include the Szabo wave equation, 6 the Caputo wave equation, 11 and the power law wave equation. 7 The Szabo and power law wave equations were developed for medical ultrasound applications, and the Caputo wave equation 11 was originally defined for applications in geophysics and then independently evaluated by Wismer as a model for attenuation and dispersion in soft tissue. 8 The time-fractional wave equations are particularly amenable to analytical methods of causality analysis, including the Paley-Wiener criterion, 12 Kramers-Kronig analysis, 13 and a time causal theory; 6 however, inconsistent conclusions are often reached with different methods, especially for power law exponents y !…”