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Shock Waves01:16

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While deriving the Doppler formula for the observed frequency of a sound wave, it is assumed that the speed of sound in the medium is greater than the source's speed through it. When this condition is breached, a shock wave occurs.
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Electromagnetic waves can travel in the vacuum as well as in matter. For example light, which is an electromagnetic wave, can travel through air, water, or glass.
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Radiating dispersive shock waves in non-local optical media.

Gennady A El1, Noel F Smyth2

  • 1Department of Mathematical Sciences , Loughborough University , Loughborough LE11 3TU, UK.

Proceedings. Mathematical, Physical, and Engineering Sciences
|April 28, 2016
PubMed
Summary

We studied nonlinear wave propagation in liquid crystals. A novel dispersive shock wave (DSW) with positive polarity was discovered, generating leading resonant radiation and exhibiting classical shock velocity.

Keywords:
dispersive shock wavenematic liquid crystalsresonant radiationundular bore

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Area of Science:

  • Nonlinear Optics
  • Condensed Matter Physics
  • Fluid Dynamics

Background:

  • Coherent light beam propagation in nematic liquid crystals follows a defocusing nonlinear Schrödinger (NLS) equation.
  • Dispersive shock waves (DSWs) are a known phenomenon in nonlinear wave propagation.
  • Standard DSW solutions of the defocusing NLS equation exhibit specific characteristics.

Purpose of the Study:

  • To analyze the step Riemann problem for light beam propagation in nematic liquid crystals.
  • To investigate the unique properties of the DSW generated in this system.
  • To develop an asymptotic theory for the nematic DSW and compare it with numerical simulations.

Main Methods:

  • Analysis of the step Riemann problem for the governing nonlinear wave equations.
  • Application of the Wentzel-Kramers-Brillouin (WKB) approximation to find the radiative wavetrain solution.
  • Derivation of an asymptotic model using a Korteweg-de Vries equation with fifth-order dispersion.
  • Direct numerical simulations for validation.

Main Results:

  • The generated DSW exhibits positive polarity, differing from standard NLS DSWs.
  • The DSW generates resonant radiation that propagates ahead of the wave.
  • The velocity of the leading soliton of the DSW is determined by the classical shock velocity.
  • The asymptotic theory, a fifth-order dispersive Korteweg-de Vries equation, accurately describes the radiation generation.

Conclusions:

  • The nematic liquid crystal system generates a unique DSW with distinct properties.
  • Resonant radiation ahead of the DSW is a key feature, explained by the derived asymptotic model.
  • The study provides a theoretical framework and numerical validation for this nonlinear wave phenomenon.