Lévy stable distribution and [0,2] power law dependence of. acoustic absorption on frequency

Transcription

1 Lév stable distribution and [,] power law dependence of acoustic absorption on frequenc W. Chen Institute of Applied Phsics and Computational Mathematics, P.O. Box 89, Division Box 6, Beijing 88, China The absorption of acoustic wave propagation in a broad variet of loss media is characterized b an empirical power law function of frequenc, ω α. It has long been noted that exponent ranges from to for diverse media. Recentl, the present author developed a fractional Laplacian wave equation to accuratel model the power law dissipation, which can be further reduced to the fractional Laplacian diffusion equation. The latter is known underling the Lév stable distribution theor. Consequentl, the parameters is found to be the Lév stabilit index, which is known bounded within <. This finding first provides a theoretical explanation of empirical observations [,]. Statisticall, the frequenc-dependent absorption can thus be understood a Lév stable process, where the parameter describes the fractal nature of attenuative media. PACS numbers: 43..Bi, 43..Hq, Bf, Cg The effect of the dissipative attenuation of acoustic wave propagation over a finite range of frequenc is tpicall characterized b a measured power law function of frequenc α = α ω, [,], () where ω denotes angular frequenc, and α and are non-negative media-dependent constants. -3. The frequenc-dependent attenuation is described b E = E e α ( ω )z. Here E represents the amplitude of an acoustic field variable such as pressure, and z is the traveling distance. It is well known that the standard mathematical modeling approach using time-space

2 derivatives of integer orders can not accuratel reflect power law function () except for two extreme cases: =,. Unfortunatel, << exponents present in most media of practical interest. For example, sediments and fractal rock laers have around,,4 and Table displas values of for different human tissues, and Fig. (reproduced from Ref. 5) shows log-log plots of absorption versus frequenc in some materials, where shear and long means shear and longitudinal waves, respectivel. YIG is the abbreviation of ttrium indium garnet, and granites and denote the two tpes of granite, respectivel. The unit decibel (db) is based on powers of (decade) to provide a relative measure of the sound intensit. The slope of the straight line is the exponent of frequenc power law of dissipation. For example, =.3 for MHz in longitudinal wave loss of bovine liver. YIG as a single crstalline material has = for both longitudinal and shear absorptions at ver high frequencies. Clearl, YIG is an ideal solid (atomic lattice) rather than soft matter (fractal macromolecules). The longitudinal wave dissipation of granite follows a linear dependence (=) on frequenc from 4 Hz to. MHz. Table. I. Tissue coefficients of frequenc-dependent power law attenuation Water 5 Fat 6 Duct cancer 6 structural tissue 6 α (db/cm/mhz ) Fig.. Data for shear and longitudinal wave loss which show power-law dependence over four decades of frequenc (taken from ref. 5).

3 I. Anomalous diffusion equation b the fractional Laplacian Among various methodologies to tackle this mathematical modeling challenge, the time derivative of fractional order has long been considered a most effective means of describing the attenuation of non-zero and non-quadratic frequenc dependenc. 8- However, it is observed from power law formula () that exponent is irrelevant to temporal frequenc ω. Instead, is found to var with media. It is therefore reasonable to think that ma underlie spatial structures of media. In fact, the temporal representation of absorption effect works under the conditions that the thermoviscous term is relativel small 4 and the interaction between two oppositel traveling sound waves can be neglected. 8 In addition, for >, the time expression of attenuation needs the initial condition of the second order derivative, which is not available in most cases. It is also impossible to express the spatial anisotropicit of attenuation via scalar time operation. Thus, a combination of spatial and temporal representation of absorption is more phsicall sound. Instead of the fractional time derivative, Chen and Holm applied the space fractional Laplacian, also known as the Riesz fractional derivative, to develop causal linear and nonlinear wave equation models being consistent with attenuations having arbitrar power law frequenc dependenc. The dissipative equation for linear isotropic media is expressed as p = c p α + c ( ) p, () where p denotes pressure, c is the small signal velocit, and ( ) represents the smmetric fractional Laplacian. - The above equation () describes both dispersion (waveform alternation with respect to frequenc) and attenuation behaviors. When =, eq. () turns out to be the thermoviscous wave equation corresponding to the squared-frequenc dependent attenuation. When =, eq. () is reduced to the standard damped wave equation reflecting frequenc-dependent attenuation. The hperbolic wave equation () can be approximated to 3

4 the generalized diffusion equation via the approach detailed in ref.. Namel, removing the left-hand side term of eq. () produces c p α + c ( ) p = And then integrating (3) with respect to time t and multipling b Laplacian diffusion equation, known as the anomalous diffusion equation c. (3), we have the fractional p + + α c ( ) p =. (4) When =, eq. (4) is the normal diffusion equation corresponding to the squared-frequenc dependent attenuation. 4 II. Lév stable distribution and [,] power law The Cauch problem of the one-dimensional anomalous diffusion equation is expressed as p + κ = p, (5) x p ( x ) = δ ( x),, p x p, (6) where κ represents the diffusion coefficient, and δ(x) is the Dirac delta function. The solution of the above equations (5) and (6) is 3 where = x,, (7) t t ( x t) w p w = dk, π iqξ ( ξ ) e W ( k ) ξ = x t. (8) W is the characteristic function of w ( ) W k κk = e, (9) (9) is also the Fourier transform of the probabilit densit function of the -stable Lév distribution. The anomalous diffusion equation is thus considered underling the Lév stable 4

5 distribution. 3,4 In the limiting case = for the standard diffusion equation, the solution is the explicit Gaussian probabilit densit function p = () 4πκt x 4κt ( x, t) e Saichev and Zaslavsk 3 pointed out that in order to satisf the positive probabilit densit function, the Lév stable index must obe p. () Namel, the -stable distribution requires the power to be positive but not greater than. 5 In particular, = corresponds to the Cauch distribution. 3. In terms of this statistical theor, the media having > power law attenuation are not statisticall stable in nature. In other words, the corresponding probabilit densit function is no longer positivel defined. It is noted that the Lév process does not include =. This means that the media obeing absolutel frequenc-independent attenuation is simpl an ideal approximation. For acoustic wave propagations, all media exhibit more or less degree of absorption dependence on frequenc. As shown in power law formula (), exponent obtained b experimental data fitting has alwas been observed within the finite scope in between and for all media. The above analsis shows that the Lév stable distribution theor provides a mathematical interpretation of empirical [,] power dependence of the absorption coefficient on the frequenc. III. Power law dissipation and fractal Rewriting the power law attenuation () as ( ω ) lnα α = () lnω clearl reveals the self-similar propert of frequenc power law dissipation. Fractal underlies self-similarit, and can thus be interpreted as the fractal dimension. On the other hand, 5

6 Mandelbrot 6 and Sato 7 note the inherent connections between the Lév stable distribution and fractals due to the inherent self-similarit of the Lév probabilit densit functions as illustrated in (7). As discussed previousl, represents the stabilit index of the Lév process, and thus is the fractal indeed. The invariance of on different frequencies (time scales) implies that depends essentiall on the space mesostructures or microstructures rather than time process. The parameter actuall represents the spatial fractal of media on diffusion process. For example, varing absorption coefficient over different human bod tissues means that the fractal characterizes the stochastic geometric propert of macromolecules of biomaterials, which dominate their phsical behaviors. Herrchen 8 points out that the self-similarit extends usuall onl over a finite range in real phsical problems. This is in agreement with man experiment observations that the power law attenuation takes effect over a finite range of finite frequenc, as illustrated in Fig.. IV. Concluding remarks Through the analsis of the fractional Laplacian models of the frequenc power law attenuation, this stud found that the exponent of the frequenc power law dissipation can be interpreted as the Lév stabilit index which are theoreticall bounded within (,). To our knowledge, this work is the first attempt to present a theoretical explanation of [,] exponent range of the power law attenuation, which has widel been observed not onl in acoustics but also in man other phsical behaviors such as vibrational damping, dielectrics, thermoviscosit, and fluid thermoviscous dissipation 4. For soft matter and non-newtonian fluids, the parameter is mostl found in between and. 6

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