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Evolution equation for nonlinear Lucassen waves, with application to a threshold phenomenon.

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  • 1Applied Research Laboratories, The University of Texas at Austin, Austin, Texas 78713-8029, USA.

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Summary

This study simplifies a complex surface wave equation, enabling efficient numerical solutions. The findings explain shock formation and propagation in elastic interfaces, crucial for understanding wave dynamics.

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Area of Science:

  • Fluid dynamics
  • Nonlinear wave propagation
  • Surface physics

Background:

  • Linear theory for wave attenuation and dispersion by Lucassen (1968).
  • Nonlinear fractional surface wave equation by Kappler et al. (2017) for elastic interfaces coupled to viscous liquids.
  • Inclusion of fractional time derivative and quadratic/cubic nonlinearity in the Kappler et al. model.

Purpose of the Study:

  • To present an integrated, first-order spatial derivative form of the Kappler et al. time-domain equation.
  • To develop an evolution equation for progressive waves that reduces computational cost.
  • To analyze waveform features, shock formation, and propagation using the simplified model.

Main Methods:

  • Derivation of an integrated, first-order spatial derivative evolution equation from the full model.
  • Numerical solution of the derived evolution equation.
  • Obtaining approximate analytical expressions for nonlinear propagation speed and attenuation.

Main Results:

  • The evolution equation successfully captures key waveform features, including shock formation and propagation, predicted by the full model.
  • Numerical solutions are computationally less expensive than the full model.
  • Analytical expressions reveal that the threshold phenomenon is linked to competing quadratic and cubic nonlinearities.

Conclusions:

  • The simplified evolution equation provides an efficient method for studying nonlinear surface waves.
  • The competition between nonlinear terms explains the observed threshold phenomenon in lipid monolayer interfaces.
  • This work offers insights into wave dynamics at fluid-elastic interfaces.