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Restrictions on wave equations for passive media.

Sverre Holm1, Martin Blomhoff Holm2

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This study explores passivity in acoustic wave equations, a stricter criterion than causality. Passive media, like viscous and relaxation models, impose constraints on relaxation modulus derivatives, ensuring slower attenuation increase and monotonic phase velocity.

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Area of Science:

  • Acoustics
  • Wave Propagation
  • Materials Science

Background:

  • Acoustic wave equations typically ensure causality.
  • Passivity is a stricter criterion than causality for linear systems.
  • Understanding passivity imposes restrictions on material properties.

Purpose of the Study:

  • To explore the consequences of requiring passivity in acoustic wave equations.
  • To establish criteria for judging the passivity of proposed wave equations.
  • To compare different viscous wave equation models based on passivity.

Main Methods:

  • Analysis of relaxation modulus and its derivatives under the passivity criterion.
  • Characterization of viscous and relaxation acoustic models as systems of springs and dampers.
  • Mathematical derivation of constraints on attenuation and phase velocity for passive media.

Main Results:

  • Passivity imposes restrictions on all derivatives of the relaxation modulus for viscous and relaxation models.
  • Attenuation in passive media increases slower than a linear function of frequency.
  • Phase velocity in passive media must increase monotonically.

Conclusions:

  • Passive acoustic media have specific, mathematically defined properties related to their relaxation modulus.
  • The derived criteria can be used to evaluate the passivity of new acoustic wave equations.
  • Comparison of viscous wave equation models highlights the importance of passivity constraints.