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On the application of Lorentz-Finsler geometry to model wave propagation

Enrique Pendás-Recondo

TL;DR

This work presents a cohesive, step-by-step framework that uses Lorentz-Finsler geometry to model wave propagation in anisotropic, time-varying media. By embedding time as a spacetime dimension and defining a Lorentz-Finsler metric $G$ with lightlike pregeodesics, the authors unify Fermat's and Huygens' principles within a single geometric scheme, enabling efficient computation of wave trajectories and fronts. The paper provides explicit methodological steps—from initial-front orthogonality to pregeodesic integration and cut-point handling—forming a practical pipeline for real-time wavefront evolution. The approach offers a flexible, accurate alternative to PDE-based wave equations, adaptable to diverse physical settings (e.g., sound, seismic, wildfires) and capable of incorporating complex, direction-dependent speed profiles through the $F$-induced indicatrix on $N$ and its spacetime extension.

Abstract

The recent increasing interest in the study of Lorentz-Finsler geometry has led to several applications to model real-world physical phenomena. Our purpose is to provide a simple, step-by-step review on how to build and implement such a geometric model to describe the propagation of a classical wave satisfying Fermat's and Huygens' principles in an anisotropic and rheonomic (time-dependent) medium. The model is based on identifying the individual wave trajectories as lightlike pregeodesics of a specific Lorentz-Finsler metric, which obey a simple ODE system and can therefore be easily computed in real time.

On the application of Lorentz-Finsler geometry to model wave propagation

TL;DR

This work presents a cohesive, step-by-step framework that uses Lorentz-Finsler geometry to model wave propagation in anisotropic, time-varying media. By embedding time as a spacetime dimension and defining a Lorentz-Finsler metric with lightlike pregeodesics, the authors unify Fermat's and Huygens' principles within a single geometric scheme, enabling efficient computation of wave trajectories and fronts. The paper provides explicit methodological steps—from initial-front orthogonality to pregeodesic integration and cut-point handling—forming a practical pipeline for real-time wavefront evolution. The approach offers a flexible, accurate alternative to PDE-based wave equations, adaptable to diverse physical settings (e.g., sound, seismic, wildfires) and capable of incorporating complex, direction-dependent speed profiles through the -induced indicatrix on and its spacetime extension.

Abstract

The recent increasing interest in the study of Lorentz-Finsler geometry has led to several applications to model real-world physical phenomena. Our purpose is to provide a simple, step-by-step review on how to build and implement such a geometric model to describe the propagation of a classical wave satisfying Fermat's and Huygens' principles in an anisotropic and rheonomic (time-dependent) medium. The model is based on identifying the individual wave trajectories as lightlike pregeodesics of a specific Lorentz-Finsler metric, which obey a simple ODE system and can therefore be easily computed in real time.
Paper Structure (20 sections, 3 theorems, 38 equations)

This paper contains 20 sections, 3 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. The setting
  3. Distance
  4. Speed profile
  5. Finsler metrics
  6. Lorentz-Finsler metrics
  7. Traveltime
  8. Time-independent case
  9. Time-dependent case
  10. Initial wavefront
  11. Geodesic equations
  12. The model
  13. The wavefront
  14. The implementation
  15. Step 1: The initial wavefront
  16. ...and 5 more sections

Key Result

Proposition 1

For any fixed $(t,x) \in \mathds{R} \times N$, the following statements are equivalent: If this holds for every $(t,x) \in \mathds{R} \times N$, then $F$ is said to be a (time-dependent) Finsler metric on $N$.See javaloyes2014 for the formal definition and properties of a Finsler metric.

Theorems & Definitions (16)

  • Example 1
  • Proposition 1
  • Remark 1
  • Example 2
  • Definition 1
  • Remark 2
  • Definition 2
  • Theorem 1
  • Remark 3
  • Remark 4
  • ...and 6 more