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Related Experiment Videos

Wave chaos in the elastic disk.

Niels Sondergaard1, Gregor Tanner

  • 1School of Mathematical Sciences, University of Nottingham, University Park, United Kingdom. niels.sondergaard@nottingham.ac.uk

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|January 7, 2003
PubMed
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A boundary integral formalism for stochastic ray tracing in billiards.

Chaos (Woodbury, N.Y.)·2015

Investigating elastic wave dynamics in disks reveals a link between wave behavior and classical ray motion. Boundary interactions introduce chaos, increasing periodic orbits and connecting wave equations to quantum graphs.

Area of Science:

  • * Physics
  • * Acoustics
  • * Wave Mechanics

Background:

  • * Explores the connection between elastic wave equations and classical ray dynamics.
  • * Focuses on the eigenfrequencies of an elastic disk with free boundaries.
  • * Examines the relationship between these frequencies and periodic rays within the disk.

Purpose of the Study:

  • * To investigate the link between elastic wave propagation and classical ray dynamics in isotropic bodies.
  • * To analyze the influence of boundary conditions on wave behavior and ray dynamics.
  • * To explore the emergence of chaotic phenomena in a classically separable system.

Main Methods:

  • * Analysis of the elastic wave equation for plane, isotropic bodies.
  • * Study of eigenfrequencies and periodic rays in a free-boundary elastic disk.

Related Experiment Videos

  • * Application of concepts from classical chaos theory and quantum graphs.
  • * Derivation of a periodic trace formula from the scattering matrix.
  • * Identification of periodic orbits via Fourier transform of spectral density.

    • Main Results:

    • * Wave mixing at boundaries introduces stochasticity into ray dynamics, leading to phenomena akin to classical chaos.
    • * An exponential increase in periodic orbits is observed.
    • * The classical problem decomposes into an integrable part and a Markov process.
    • * The wave equation maps to a quantum graph in the high-frequency limit.
    • * A periodic trace formula is derived, linking spectral properties to periodic orbits.

    Conclusions:

    • * The study establishes a connection between wave phenomena and chaotic ray dynamics in elastic disks.
    • * The findings suggest that complex wave behavior can emerge from seemingly simple systems due to boundary interactions.
    • * The mapping to quantum graphs provides new avenues for understanding wave behavior and level statistics.