# Quasi-normal modes in a symmetric triangular barrier

## TL;DR

This paper analytically derives quasi-normal modes for a scalar wave in a symmetric triangular barrier potential, demonstrating their relevance in wormhole geometries and providing numerical and approximate solutions.

## Contribution

It presents an exact solution for QNMs in a triangular barrier potential and connects these results to wormhole physics and scalar perturbations.

## Key findings

- Exact QNM frequencies obtained using Airy functions.
- Numerical wave profiles illustrate quasinormal ringing.
- Approximate modeling of wormhole perturbations with triangular barriers.

## Abstract

Quasi-normal modes (QNMs) of the massless scalar wave in $1+1$ dimensions are obtained for a symmetric, finite, triangular barrier potential. This problem is exactly solvable, with Airy functions involved in the solutions. Before obtaining the QNMs, we demonstrate how such a triangular barrier may arise in the context of scalar wave propagation in a tailor-made wormhole geometry. Thereafter, the Ferrari-Mashhoon idea is used to show how bound states in a well potential may be used to find the QNMs in a corresponding barrier potential. The bound state condition in the exactly solvable triangular well and the transformed condition for finding the QNMs are written down. Real bound state energies and complex QNMs are found by solving the respective transcendental equations. Numerical integration of the wave equation yields the time domain profiles for scalar waves propagating in this wormhole geometry which illustrate the quasinormal ringing. Estimates relating the size of the wormhole throat (in units of solar mass) with the QNM frequencies are stated and discussed. Finally, we show how the effective potential and the QNMs for scalar perturbations of the Ellis--Bronnikov wormhole spacetime can be reasonably well--approximated using a properly parametrised triangular barrier.