# Invisibility and perfect reflectivity in waveguides with finite length branches

## TL;DR

This paper analyzes waveguide geometries with finite branches to achieve invisibility, perfect reflection, or zero reflection by studying the behavior of scattering coefficients as the branch length varies.

## Contribution

It introduces a method to design waveguide geometries that realize invisibility, perfect reflection, or zero reflection by controlling the branch length and geometry.

## Key findings

- Conditions for non-reflectivity ($ ext{Re}= ext{Im}=0$) are established.
- Conditions for perfect reflectivity ($| ext{Re}|=1$, $ ext{Im}=0$) are identified.
- Conditions for perfect invisibility ($ ext{Re}=0$, $| ext{Im}|=1$) are demonstrated.

## Abstract

We consider a time-harmonic wave problem, appearing for example in water-waves and in acoustics, in a setting such that the analysis reduces to the study of a 2D waveguide problem with a Neumann boundary condition. The geometry is symmetric with respect to an axis orthogonal to the direction of propagation of waves. Moreover, the waveguide contains one branch of finite length. We analyse the behaviour of the complex scattering coefficients $\mathcal{R}$, $\mathcal{T}$ as the length of the branch increases and we show how to design geometries where non reflectivity ($\mathcal{R}=0$, $|\mathcal{T}|=1$), perfect reflectivity ($|\mathcal{R}|=1$, $\mathcal{T}=0$) or perfect invisibility ($\mathcal{R}=0$, $\mathcal{T}=1$) hold. Numerical experiments illustrate the different results.