Titlebook: Waves with Power-Law Attenuation; Sverre Holm Book 2019 Springer Nature Switzerland AG 2019 Power laws acoustics.Linear viscoelasticity.El

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, extended problem. The central question, as ever, was whether a general rapprochement could be reached that would prevent the outbreak of a European war, or, at least, limit British involvement in such a conflict. The policy of appeasement remained associated more with peace and reconciliation rath

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Introduction, and the various mechanisms that can cause such attenuation. Then attenuation will follow: .where . is the attenuation (the negative imaginary part of the wave number ., hence, the index), . is angular frequency, and . and . are constants. The absolute value is used to ensure that attenuation never

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Absorption Mechanisms and Physical Constraintsch smaller than the wavelength, often at the molecular level. This is especially so in the acoustics case. In Sect.?. specific examples of processes were given due to ., ., and . for seawater as well as . and . for air. This chapter begins with a more detailed account of these processes starting wit

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Power-Law Wave Equations from Constitutive Equationsnd Zener models (.) and (.) as well as the relaxation responses of the Maxwell and Zener models ((.) and (.)). But in many complex media this model is too simple, and power laws are observed instead, which may both be in time-domain responses and in the frequency-domain. As the Fourier transform of

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Phenomenological Power-Law Wave Equationse consideration for time domain properties. First wave equations which are found by manipulating the common wave equations will be discussed. This is done by replacing ordinary derivatives in time or space with fractional ones. In some cases, this will give the same solutions as those found from fra

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Introduction, and the various mechanisms that can cause such attenuation. Then attenuation will follow: .where . is the attenuation (the negative imaginary part of the wave number ., hence, the index), . is angular frequency, and . and . are constants. The absolute value is used to ensure that attenuation never

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Classical Wave Equations the viscous and the relaxation ones. It is shown that the viscous model derives from the Kelvin–Voigt spring–damper system, and that the relaxation model is based on the standard linear solid or Zener model. The multiple-relaxation model for seawater and air are also shown to be based on the Maxwel