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Classical Wave Equations

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Waves with Power-Law Attenuation

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Abstract

The aim of this chapter is to describe the conventional acoustic models in the framework of linear elasticity. The two main attenuation mechanisms are 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 Maxwell–Wiechert model, which is a generalization of the Zener model. This establishes the foundation for later generalization to fractional versions of the linear viscoelastic models.

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Notes

  1. 1.

    The opposite naming convention, with \(\tau _\varepsilon \) and \(\tau _\sigma \) exchanged, is also used. Examples are Zener (1948), Mainardi (1994), Holm and Sinkus (2010), Holm and Näsholm (2014), but here the definitions follow that of Mainardi (2010) as then the step response in \(\sigma \) will be described by \(\tau _\sigma \) and the same for \(\varepsilon \) and \(\tau _\varepsilon \), see (3.45) and (3.47).

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  1. Department of Informatics, University of Oslo, Oslo, Norway

    Sverre Holm

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  1. Sverre Holm
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Holm, S. (2019). Classical Wave Equations. In: Waves with Power-Law Attenuation. Springer, Cham. https://doi.org/10.1007/978-3-030-14927-7_2

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