Power-law attenuation in acoustic and isotropic anelastic media

Abstract

S U M M A R YPower attenuation laws were originally introduced for phenomenological descriptions of attenuation in a large variety of materials. More recently they have been applied in viscoelastic inversion. Power-law models can be readily expressed in terms of scalar acoustic partial integro-differential equations. The restrictions on the exponent of the power law imposed by the assumptions of causality and dissipativity are discussed. An extension of power-law attenuation models from acoustic to isotropic anelastic media is proposed. The guiding principle is that the attenuation of longitudinal and transverse waves satisfies the power law. Restrictions imposed by thermodynamics and well-posedness considerations are discussed.

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

“…Power-law attenuation has been incorporated in the time-domain via FPDEs, which add loss to the wave equation with a time-fractional derivative [4][5][6][7] or a space-fractional derivative. 8,9 Unlike loss models that utilize integer-order derivatives, 10,11 these FPDEs support power-law attenuation coefficients for a range of noninteger exponents y.…”

Effects of Newtonian viscosity and relaxation on linear viscoelastic wave propagation

“…We shall change the variable k: k = −iκ(p). The solution of equation (10) can be expressed in the following form κ(p)/p = ρ 1/2 / N p + pG(p) 1/2 (11) with the square root chosen so that Re κ(p) ≥ 0 for Re p > 0. The function κ(p) is known as the complex wavenumber function [2].…”

Stability for the damped wave equation with neutral delay

“…They are known as "Neutral Delay Differential Equations" (NDDEs) [2], [6][7][8][9], [11], [15], [16], [20], [22], [30], [37][38][39][40][41][42][43][44][45]. We can find them in the study of heat exchanges, control theory, biology, biosciences, in electric networks of high speed computers where the lossless transmission lines are used to interconnect switching circuits and in population ecology [1], [15][16][17][18][19][20], [23], [29], [41][42][43].…”

“…Here △ denotes the usual Laplacian. Several researchers have shown that even an extremely "small" delay may generate instability, oscillations and chaos [1], [3][4][5], [12], [19]. This, of course, leads to a deficiency and abnormality causing anxiety to the user.…”