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

TL;DR

This paper extends the classical d'Alembert formula to solve wave scattering problems in one dimension, including inverse problems, using the Fokas method for media with constant and piecewise constant refractive indices, and provides an exact reconstruction method.

Contribution

It introduces a novel extension of the d'Alembert formula for wave scattering and develops an exact reconstruction method for inverse problems using the Fokas method.

Findings

Derived an analytical solution for direct scattering in constant media

Extended the methodology to piecewise constant media for inverse problems

Provided an exact reconstruction method applicable to full and phaseless data

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.