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This study provides a more accessible derivation of the Green's function parabolic equation (GFPE) algorithm. It simplifies the underlying physics and mathematics for engineers and physicists using familiar methods for acoustic propagation analysis.

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Area of Science:

  • Computational physics
  • Acoustic wave propagation
  • Mathematical modeling

Background:

  • The Green's function parabolic equation (GFPE) algorithm is crucial for modeling acoustic propagation.
  • Existing derivations of the GFPE algorithm can be mathematically complex for some practitioners.
  • A clear understanding of the GFPE's underlying physics and mathematics is beneficial.

Purpose of the Study:

  • To present a more accessible derivation of the Green's function parabolic equation (GFPE) algorithm.
  • To clarify the mathematical and physical principles behind the GFPE algorithm.
  • To provide a foundation for understanding one-way acoustic propagation solutions.

Main Methods:

  • Utilized the separation of variables method for solving partial differential equations.
  • Derived analytic expressions for horizontal and vertical eigenfunctions (sinusoids plus surface wave).
  • Employed eigenfunction expansion to construct a one-way propagation solution and generalize to the GFPE algorithm.

Main Results:

  • Developed a step-by-step derivation of the GFPE algorithm using accessible mathematical techniques.
  • Obtained simplified, analytic expressions for eigenfunctions.
  • Demonstrated how eigenfunction expansion provides concrete meaning to abstract operator solutions.

Conclusions:

  • The presented derivation enhances the accessibility of the GFPE algorithm for engineers and physicists.
  • Eigenfunction expansion offers a direct and concise method for obtaining the GFPE algorithm.
  • This work facilitates a deeper understanding of one-way acoustic propagation modeling.