Modal analysis of wave propagation in dispersive media

TL;DR

This paper introduces a new method to eliminate branch points in wave integral expressions in dispersive media, enabling a modal expansion that improves understanding and analytical description of wave transients like Sommerfeld and Brillouin precursors.

Contribution

A novel approach to remove critical branches in wave integrals, allowing for a clear modal expansion and enhanced analytical modeling of wave propagation in dispersive media.

Findings

Elimination of branch points in wave integral expressions.

Improved analytical description of Sommerfeld and Brillouin precursors.

Enhanced understanding of transient wave components.

Abstract

Surveys on wave propagation in dispersive media have been limited since the pioneering work of Sommerfeld [Ann. Phys. 349, 177 (1914)] by the presence of branches in the integral expression of the wave function. In this article, a method is proposed to eliminate these critical branches and hence to establish a modal expansion of the time-dependent wave function. The different components of the transient waves are physically interpreted as the contributions of distinct sets of modes and characterized accordingly. Then, the modal expansion is used to derive a modified analytical expression of the Sommerfeld precursor improving significantly the description of the amplitude and the oscillating period up to the arrival of the Brillouin precursor. The proposed method and results apply to all waves governed by the Helmholtz equations.