# Explicit inversion formulas for the two-dimensional wave equation from Neumann traces

## TL;DR

This paper develops explicit inversion formulas to recover initial data of the 2D wave equation from Neumann boundary measurements, providing exact solutions for circular and elliptical domains with numerical validation.

## Contribution

It introduces new explicit inversion formulas for the 2D wave equation from Neumann data, including exact solutions for circular and elliptical geometries.

## Key findings

- Derived explicit back-projection inversion formulas.
- Achieved exact recovery for circular and elliptical domains.
- Numerical results demonstrate accuracy and stability.

## Abstract

In this article we study the problem of recovering the initial data of the two-dimensional wave equation from Neumann measurements on a convex domain with smooth boundary in the plane. We derive an explicit inversion formula of a so-called back-projection type and deduce exact inversion formulas for circular and elliptical domains. In addition, for circular domains, we show that the initial data can also be recovered from any linear combination of its solution and its normal derivative on the boundary. Numerical results of our implementation of the derived inversion formulas are presented demonstrating their accuracy and stability.