# Exact analytical solutions in inhomogeneous magnetic fields for linear asteroseismic waves

## TL;DR

This paper derives exact analytical solutions for linear asteroseismic waves in an inhomogeneous magnetic field within an isothermal, stratified medium, revealing how magnetic inhomogeneity affects wave behavior and potential stellar interior diagnostics.

## Contribution

It provides new exact solutions for wave propagation in stratified magnetic environments, highlighting the impact of magnetic inhomogeneity on eigenfrequencies and mode structures.

## Key findings

- Eigenfrequencies significantly affected by Alfvén speed variations
- Wave reflection occurs in regions with larger Alfvén speed
- Mode structures show increased oscillations and leakage in inhomogeneous fields

## Abstract

We solve for waves in an isothermal, stratified medium with a magnetic field that points along a direction perpendicular to that of gravity and varies exponentially in the direction of gravity. We find exact analytical solutions for two different cases: (a) waves propagating along the direction of the magnetic field and (b) waves propagating along the direction of the gravity. In each case, we find solutions in terms of either the hypergeometric functions or their confluent cousins. We solve the resultant transcendental dispersion relation numerically. The changes in the eigenfrequencies due to a spatially inhomogeneous Alfv\'en wave speed are significant -- there are more oscillations or leaking of nodes of the eigenfunctions in regions of lower Alfv\'en speed and wave reflection from regions of larger Alfv\'en speed. Such changes in the dispersion relation and the mode structures may allow detection of magnetic fields buried in the stellar interior.