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Wave scattering in 1D: D'Alembert-type representations and a reconstruction method

Konstantinos Kalimeris, Leonidas Mindrinos

TL;DR

This work studies the 1-D wave equation on the half-line with a layered medium of total length $L$, addressing direct scattering and inverse reconstruction from backscattered data at $x=L+D$. Using the Fokas unified transform, it derives a d'Alembert-type representation that extends the classical solution to piecewise constant refractive indices, producing finite-sum formulas rather than infinite series. For the inverse problem, it presents exact reconstruction methods for single-, double-, and multi-layer configurations with both full data and phaseless data, recovering layer speeds $\{c_j\}$ and thicknesses $\{\ell_j\}$ from backscattering peaks, with the total length constraint providing a disambiguation mechanism in the phaseless case. Major reflections from interfaces yield a tractable peak-based reconstruction strategy, and the approach is validated by numerical examples that confirm the analytic representations and demonstrate potential extensions to variable-index media and higher dimensions.

Abstract

We derive the extension of the classical d'Alembert formula for the wave equation, which provides the analytical solution for the direct scattering problem for a medium with constant refractive index; this is achieved by employing results obtained via the Fokas method. This methodology is further extended to a medium with piecewise constant refractive index, providing the apparatus for the solution of the associated inverse scattering problem. Hence, we provide an exact reconstruction method which is valid for both full and phaseless data.

Wave scattering in 1D: D'Alembert-type representations and a reconstruction method

TL;DR

This work studies the 1-D wave equation on the half-line with a layered medium of total length , addressing direct scattering and inverse reconstruction from backscattered data at . Using the Fokas unified transform, it derives a d'Alembert-type representation that extends the classical solution to piecewise constant refractive indices, producing finite-sum formulas rather than infinite series. For the inverse problem, it presents exact reconstruction methods for single-, double-, and multi-layer configurations with both full data and phaseless data, recovering layer speeds and thicknesses from backscattering peaks, with the total length constraint providing a disambiguation mechanism in the phaseless case. Major reflections from interfaces yield a tractable peak-based reconstruction strategy, and the approach is validated by numerical examples that confirm the analytic representations and demonstrate potential extensions to variable-index media and higher dimensions.

Abstract

We derive the extension of the classical d'Alembert formula for the wave equation, which provides the analytical solution for the direct scattering problem for a medium with constant refractive index; this is achieved by employing results obtained via the Fokas method. This methodology is further extended to a medium with piecewise constant refractive index, providing the apparatus for the solution of the associated inverse scattering problem. Hence, we provide an exact reconstruction method which is valid for both full and phaseless data.
Paper Structure (21 sections, 3 theorems, 85 equations, 5 figures, 2 tables)

This paper contains 21 sections, 3 theorems, 85 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The direct scattering problem
  3. Single-layered medium
  4. Delta-like short pulse
  5. Double-layered medium
  6. The inverse scattering problem
  7. Single-layered medium
  8. Full-data
  9. Phaseless-data
  10. Double-layered medium
  11. Full-data
  12. Phaseless-data
  13. Multi-layered medium
  14. Full-data
  15. Phaseless-data
  16. ...and 6 more sections

Key Result

Theorem 2.1

Let $U_0$ be supported in $(L,\, \infty),$ then the solution of final_eq1 is given by under the convention that if $0\leq t<2L/c_1,$ then the above sum yields no terms, thus vanishes.

Figures (5)

  • Figure 1: The iterative scheme for a $3-$layered medium with length $L$. If all solutions are in the $(0,1)-$interval, we obtain 4 sequences of possible wave speeds. In Step 4, the output of the algorithm is the sequence whose elements satisfy $\sum_{j=1}^3 (t_{j+1}-t_j) c_j = 2L$.
  • Figure 2: The function $W$ given by \ref{['sol_r']} for the first example (left) and its projection in the $x-t$ plane (right). The plane (in red) identifies the boundary at $x=3.$
  • Figure 3: The function $W$ given by \ref{['sol_r']} for the second example (left) and its projection in the $x-t$ plane (right). The plane (in red) identifies the boundary at $x=2.$
  • Figure 4: The cross-section of $u$ at $x=8,$ corresponding to the data $m$ (solid blue line), for the first example (left) and for the second example (right). For the first example this analytical solution is also compared with the numerical solution using FDM (red dotted line).
  • Figure 5: The function $u(5,t),$ for $t\in[0,30]$ of the third example. In the left picture the minor peaks appear after the major peaks, whereas in the right the minor peaks appear in between the major peaks; the red arrows point at the minor peaks.

Theorems & Definitions (9)

  • Theorem 2.1
  • proof
  • Remark 2.2
  • Proposition 2.3
  • Remark 2.4
  • Remark 2.5
  • Proposition 2.6
  • proof
  • Remark 3.1