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Acoustical Klein-Gordon equation: a time-independent perturbation analysis.

Barbara J Forbes1, E Roy Pike

  • 1Phonologica, PO Box 43925, London NW2 1DJ, United Kingdom.

Physical Review Letters
|August 25, 2004
PubMed
Summary

Rayleigh's 1878 acoustical duct analysis is refined by considering energy density fluctuations. Perturbations in potential energy, not constant densities, accurately define eigenvalue shifts in acoustical systems.

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Area of Science:

  • Acoustics
  • Fluid Dynamics
  • Mathematical Physics

Background:

  • Classical perturbation analysis of acoustical ducts, established by Rayleigh in 1878, assumes constant energy densities during cross-section changes.
  • This assumption is challenged by significant fluctuations in potential and kinetic energy densities relative to the acoustical wave function.

Purpose of the Study:

  • To re-evaluate the perturbation analysis of ideal acoustical ducts.
  • To investigate the role of energy density fluctuations in eigenvalue shifts.
  • To establish a more accurate method for calculating eigenvalue shifts in acoustical systems.

Main Methods:

  • Revisiting Rayleigh's perturbation analysis framework.
  • Analyzing the acoustical Klein-Gordon equation and its wave function, Psi(x,t).

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  • Examining the relationship between the square of the time-independent eigenfunction, psi(2)(x), and potential energy per unit length.

    • Main Results:

    • Demonstrated that potential and kinetic energy densities fluctuate significantly.
    • Showed that the square of the time-independent eigenfunction, psi(2)(x), is proportional to the potential energy per unit length.
    • Identified that perturbations in potential energy accurately define eigenvalue shifts.

    Conclusions:

    • The assumption of constant energy densities in classical acoustical duct analysis is inaccurate.
    • Perturbations in potential energy are the critical factor in determining eigenvalue shifts.
    • This revised understanding offers a more precise approach to acoustical system analysis.