# A Stable Boundary Integral Formulation of an Acoustic Wave Transmission Problem with Mixed Boundary Conditions

## TL;DR

This paper develops a stable boundary integral formulation for acoustic wave transmission problems with mixed boundary conditions, using space-time retarded integral equations and novel single-trace spaces, ensuring mathematical stability.

## Contribution

It introduces a direct boundary integral approach with a new space-time formulation that handles mixed boundary conditions and transmission interfaces independently.

## Key findings

- Proves continuity and coercivity of the formulation.
- Defines single-trace spaces for mixed boundary conditions.
- Employs operational calculus in the Laplace domain.

## Abstract

In this paper, we consider an acoustic wave transmission problem with mixed boundary conditions of Dirichlet, Neumann, and impedance type. The transmission interfaces may join the domain boundary in a general way independent of the location of the boundary conditions. We will derive a formulation as a \textit{direct}, \textit{space-time retarded boundary integral equation}, where both Cauchy data are kept as unknowns on the impedance part of the boundary. This requires the definition of single-trace spaces which incorporate homogeneous Dirichlet and Neumann conditions on the corresponding parts on the boundary. We prove the continuity and coercivity of the formulation by employing the technique of operational calculus in the Laplace domain.