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Acoustic attenuation is a measure of the energy loss of sound propagation in media. Most media have viscosity, and are therefore not ideal media. When sound propagates in such media, there is always thermal consumption of energy caused by viscosity. For inhomogeneous media, besides media viscosity, acoustic scattering is another main reason for removal of acoustic energy. Acoustic attenuation in a lossy medium plays an important role in many scientific researches and engineering fields, such as medical ultrasonography, vibration and noise reduction. ==Power-law frequency-dependent acoustic attenuation== Many experimental and field measurements show that the acoustic attenuation coefficient of a wide range of viscoelastic materials, such as soft tissue, polymers, soil and porous rock, can be expressed as the following power law with respect to frequency:〔Szabo T. L., and Wu J., 2000, “A model for longitudinal and shear wave propagation in viscoelastic media,” J. Acoust. Soc. Am., 107(5), pp. 2437-2446.〕〔Szabo T. L., 1994, “Time domain wave equations for lossy media obeying a frequency power law,” J. Acoust. Soc. Am., 96(1), pp. 491-500.〕〔Chen W., and Holm S., 2003, “Modified Szabo’s wave equation models for lossy media obeying frequency power law,” J. Acoust. Soc. Am., 114(5), pp. 2570-2574.〕 : where is the angular frequency, ''P'' the pressure, the wave propagation distance, the attenuation coefficient, and frequency dependent exponent are real non-negative material parameters obtained by fitting experimental data and the value of ranges from 0 to 2. Acoustic attenuation in water, many metals and crystalline materials are frequency-squared dependent, namely . In contrast, it is widely noted that the frequency dependent exponent of viscoelastic materials is between 0 and 2.〔〔〔〔Carcione J. M., Cavallini F., Mainardi F., and Hanyga A., 2002, “Time-domain Modeling of Constant-Q Seismic Waves Using Fractional Derivatives,” Pure appl. geophys., 159, pp. 1719-1736.〕〔D’astrous F. T., and Foster F. S., 1986, “Frequency dependence of ultrasound attenuation and backscatter in breast tissue,” Ultrasound Med. Biol., 12(10), pp. 795-808.〕 For example, the exponent of sediment, soil and rock is about 1, and the exponent of most soft tissues is between 1 and 2.〔〔〔〔〔 The classical dissipative acoustic wave propagation equations are confined to the frequency-independent and frequency-squared dependent attenuation, such as damped wave equation and approximate thermoviscous wave equation. In recent decades, increasing attention and efforts are focused on developing accurate models to describe general power law frequency-dependent acoustic attenuation.〔〔〔Chen W., and Holm S., 2004, “Fractional Laplacian time-space models for linear and nonlinear lossy media exhibiting arbitrary frequency power-law dependency,” The Journal of the Acoustical Society of America, 115(4), pp. 1424-1430.〕〔Holm S., and Näsholm S. P., 2011, "A causal and fractional all-frequency wave equation for lossy media," The Journal of the Acoustical Society of America, 130(4), pp. 2195-2201.〕〔Pritz T., 2004, “Frequency power law of material damping,” Applied Acoustics, 65, pp. 1027-1036.〕〔Waters K. R., Mobley J., and Miller J. G., 2005, “Causality-Imposed (Kramers-Kronig) Relationships Between Attenuation and Dispersion,” IEEE Trans. Ultra. Ferro. Freq. Contr., 52(5), pp. 822-833.〕〔Nachman A. I., Smith J. F., and Waag R. C., 1990, “An equation for acoustic propagation in inhomogeneous media with relaxation losses,” J. Acoust. Soc. Am., 88(3), pp. 1584-1595.〕〔Caputo M., and Mainardi F., 1971, “A new dissipation model based on memory mechanism,” Pure and Applied Geophysics, 91(1), pp. 134-147.〕 Most of these recent frequency-dependent models are established via the analysis of the complex wave number and are then extended to transient wave propagation.〔Thomas L. Szabo, 2004, Diagnostic ultrasound imaging, Elsevier Academic Press.〕 The multiple relaxation model considers the power law viscosity underlying different molecular relaxation processes.〔 Szabo〔 proposed a time convolution integral dissipative acoustic wave equation. On the other hand, acoustic wave equations based on fractional derivative viscoelastic models are applied to describe the power law frequency dependent acoustic attenuation.〔 Chen and Holm proposed the positive fractional derivative modified Szabo's wave equation〔 and the fractional Laplacian wave equation.〔 See 〔Holm S., Näsholm, S. P., "Comparison of Fractional Wave Equations for Power Law Attenuation in Ultrasound and Elastography," Ultrasound Med. Biol., 40(4), pp. 695-703, DOI: 10.1016/j.ultrasmedbio.2013.09.033 (Link to e-print )〕 for a recent paper which compares fractional wave equations which model power-law attenuation. The phenomenon of attenuation obeying a frequency power-law may be described using a causal wave equation, derived from a fractional constitutive equation between stress and strain. This wave equation incorporates fractional time derivatives: 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「acoustic attenuation」の詳細全文を読む スポンサード リンク
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