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Acoustic pulse propagation in a non-ideal shallow-water model

Aleksandr Kaplun, Boris Katsnelson

TL;DR

This work develops a non-ideal shallow-water acoustic framework by formulating an $\varepsilon$-pseudodifferential operator that separates fast vertical and slow horizontal dynamics, and by applying a WKB/Maslov ansatz to yield single-mode eikonal and transport equations within a Hamiltonian structure. By allowing non-self-adjoint operators, the model captures bottom leakage and dissipative effects, unifying ray-based and modal descriptions with a consistent phase-space evolution for amplitude and phase. The vertical-mode analysis, including eigenvalue problems and adjoint relations, connects the vertical structure to horizontal propagation through a real-valued Hamiltonian in the dissipative limit and a complex extension for loss. Analytical and numerical examples illustrate how boundary conditions (Dirichlet, Neumann, and partial reflection) modify fronts, amplitudes, and phase, highlighting the framework’s potential to enhance realism in coastal acoustics and provide a bridge between semiclassical theory and oceanographic applications.

Abstract

This study develops a theoretical framework for modeling acoustic pulse propagation in a non-ideal shallow-water waveguide. We derive an ε-pseudodifferential operator (ε-PDO) formulation from the general three-dimensional wave equation, that accounts for vertical stratification, bottom interaction, and slow horizontal inhomogeneity. Using the operator separation of variables method and the WKB-ansatz, we obtain single-mode equations describing the evolution of amplitude and phase along rays. The approach incorporates non-self-adjoint operators to model energy leakage through the bottom and introduces a Hamiltonian formalism for eikonal and transport equations, enabling the computation of amplitude, time, and phase fronts. Analytical and numerical examples are provided for different boundary conditions, including Neumann (ideal), self-adjoint, and partially reflecting interfaces. The results extend previous semiclassical and ray-based theories of wave propagation by including dissipative effects and improving the physical realism of shallow-water acoustic modeling.

Acoustic pulse propagation in a non-ideal shallow-water model

TL;DR

This work develops a non-ideal shallow-water acoustic framework by formulating an -pseudodifferential operator that separates fast vertical and slow horizontal dynamics, and by applying a WKB/Maslov ansatz to yield single-mode eikonal and transport equations within a Hamiltonian structure. By allowing non-self-adjoint operators, the model captures bottom leakage and dissipative effects, unifying ray-based and modal descriptions with a consistent phase-space evolution for amplitude and phase. The vertical-mode analysis, including eigenvalue problems and adjoint relations, connects the vertical structure to horizontal propagation through a real-valued Hamiltonian in the dissipative limit and a complex extension for loss. Analytical and numerical examples illustrate how boundary conditions (Dirichlet, Neumann, and partial reflection) modify fronts, amplitudes, and phase, highlighting the framework’s potential to enhance realism in coastal acoustics and provide a bridge between semiclassical theory and oceanographic applications.

Abstract

This study develops a theoretical framework for modeling acoustic pulse propagation in a non-ideal shallow-water waveguide. We derive an ε-pseudodifferential operator (ε-PDO) formulation from the general three-dimensional wave equation, that accounts for vertical stratification, bottom interaction, and slow horizontal inhomogeneity. Using the operator separation of variables method and the WKB-ansatz, we obtain single-mode equations describing the evolution of amplitude and phase along rays. The approach incorporates non-self-adjoint operators to model energy leakage through the bottom and introduces a Hamiltonian formalism for eikonal and transport equations, enabling the computation of amplitude, time, and phase fronts. Analytical and numerical examples are provided for different boundary conditions, including Neumann (ideal), self-adjoint, and partially reflecting interfaces. The results extend previous semiclassical and ray-based theories of wave propagation by including dissipative effects and improving the physical realism of shallow-water acoustic modeling.

Paper Structure

This paper contains 27 sections, 113 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Wave equation
  3. Initial problem
  4. $\varepsilon-$PDO formulation
  5. WKB-ansatz for single mode
  6. $\varepsilon$-PDO and WKB-ansatz
  7. Non-dissipative case
  8. Real eigenvalues
  9. Equation for WKB-ansatz
  10. Vertical modes
  11. Eigenvalue-dependent problem
  12. Operators $\mathcal{L}(p_{\tau},\vec{r})$
  13. Operators $\mathcal{L}^*(p,\vec{r})$
  14. Relation between operators
  15. Hamiltonian dynamics
  16. ...and 12 more sections

Figures (5)

  • Figure 1: Medium properties
  • Figure 2: Amplitude fronts, $l = 1$, $\mu_1 = 0$, $\mu_2\in[-\pi,\pi]$, $\bm{\tau} = 5$
  • Figure 3: Amplitude fronts, $\alpha = \frac{1}{2}$, $l = 1$, $\mu_2\in[-\pi,\pi]$, $\bm{\tau} = 5$
  • Figure 4: Amplitude fronts, $\alpha = \frac{1}{2}$, $l = 1$, $\mu_2\in\left[\frac{5}{12}\pi ,\frac{7}{12}\pi \right]$, $\bm{\tau} = 10$
  • Figure 5: Time fronts, $\alpha = \frac{1}{2}$, $l = 1$, $\mu_2\in\left[\frac{5}{12}\pi ,\frac{7}{12}\pi \right]$, $\bm{\tau} = 10$