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Rheologica Acta

, Volume 58, Issue 1–2, pp 21–28 | Cite as

On the Kramers-Kronig relations

  • José M. Carcione
  • Fabio Cavallini
  • Jing BaEmail author
  • Wei Cheng
  • Ayman N. Qadrouh
Original Contribution
  • 102 Downloads

Abstract

We provide a new derivation of the Kramers-Kronig relations on the basis of the Sokhotski-Plemelj equation with detailed mathematical justifications. The relations hold for a causal function, whose Fourier transform is regular (holomorphic) and square-integrable. This implies analyticity in the lower complex plane and a Fourier transform that vanishes at the high-frequency limit. In viscoelasticity, we show that the complex and frequency-dependent modulus describing the stiffness does not satisfy the relation but the modulus minus its high-frequency value does it. This is due to the fact that despite its causality, the modulus is not square-integrable due to a non-null instantaneous response. The relations are obtained in addition for the wave velocity and attenuation factor. The Zener, Maxwell, and Kelvin-Voigt viscoelastic models illustrate these properties. We verify the Kramers-Kronig relations on experimental data of sound attenuation in seabottoms sediments.

Keywords

Kramers-Kronig relations Sokhotski-Plemelj equation Causality Viscoelasticity Waves Zener model 

Notes

Acknowledgements

This work is supported by the Specially-Appoin- ted Professor Program of Jiangsu Province, China, the Cultivation Program of “111” Plan of China (BC2018019) and the Fundamental Research Funds for the Central Universities, China.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • José M. Carcione
    • 1
    • 2
  • Fabio Cavallini
    • 1
  • Jing Ba
    • 2
    Email author
  • Wei Cheng
    • 2
  • Ayman N. Qadrouh
    • 3
  1. 1.Istituto Nazionale di Oceanografia e di Geofisica Sperimentale (OGS)SgonicoItaly
  2. 2.School of Earth Sciences and EngineeringHohai UniversityNanjingChina
  3. 3.SAC - KACSTRiyadhSaudi Arabia

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