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


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.


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



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.


  1. Bracewell R (1965) The Fourier transform and its applications. McGraw-Hill Book CoGoogle Scholar
  2. Carcione JM (1999) On energy definition in electromagnetism: an analogy with viscoelasticity. J Acous Soc Am 105(2):626–632CrossRefGoogle Scholar
  3. Carcione JM (2014) Wave fields in real media. Theory and numerical simulation of wave propagation in anisotropic, anelastic, porous and electromagnetic media. Elsevier. (Third edition, extended and revised)Google Scholar
  4. Frisch M (2009) ‘The most sacred tenet’? Causal reasoning in physics. Br J Philos Sci 60(3):459–474CrossRefGoogle Scholar
  5. Golden JM, Graham GAC (1988) Boundary value problems in linear viscoelasticity. Springer, BerlinCrossRefGoogle Scholar
  6. Kramers HA (1927) La diffusion de la lumiere par les atomes. Atti Congr Intern Fisica, Como 2:545–557Google Scholar
  7. Kronig R de L (1926) On the theory of the dispersion of X-rays. J Opt Soc Am 12:547–557CrossRefGoogle Scholar
  8. Labuda C, Labuda I (2014) On the mathematics underlying dispersion relations. Eur Phys J H,
  9. Liu HP, Anderson DL, Kanamori H (1976) Velocity dispersion due to anelasticity; implications for seismology and mantle composition. Geophys J Roy Astr Soc 47:41–58CrossRefGoogle Scholar
  10. Nussenzveig HM (1960) Causality an dispersion relations for fixed momentum transfer. Physics 26:209–229Google Scholar
  11. Nussenzveig HM (1972). In: Bellman R (ed) Causality and dispersion relations, mathematics in science and engineering, vol 95. Academic Press, New YorkGoogle Scholar
  12. Plemelj J (1908) Riemannsche Funktionenscharen mit gegebener Monodromiegruppe. Monatshefte für Mathematik und Physik 19:211–246CrossRefGoogle Scholar
  13. Pritz T (1999) Verification of local Kramers-Kronig relations for complex modulus by means of fractional derivative model. J Sound Vib 228:1145–1165CrossRefGoogle Scholar
  14. Sokhotskii YW (1873) On definite integrals and functions used in series expansions. St. PetersburgGoogle Scholar
  15. Stastna J, De Kee D, Powley MB (1985) Complex viscosity as a generalized response function. J Rheol 29:457–469CrossRefGoogle Scholar
  16. Toksöz MN, Johnston DH (eds) (1981) Seismic wave attenuation. Geophysical Reprint Series, TulsaGoogle Scholar
  17. Wang Y (2007) Seismic inverse Q filtering. Blackwell PubGoogle Scholar
  18. Zener C (1948) Elasticity and anelasticity of metals. University of Chicago, ChicagoGoogle Scholar
  19. Zhou J-X, Zhang X-Z, Knobles DP (2009) Low-frequency geoacoustic model for the effective properties of sandy seabottoms. J Acoust Soc Amer 125:2847–2866CrossRefGoogle Scholar

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

Personalised recommendations