# On Christoffel roots for nondetached slowness surfaces

## TL;DR

This paper investigates the properties of Christoffel roots for nondetached $qP$ slowness surfaces in transversely isotropic media, revealing differences from detached surfaces in wavefront correspondence and surface smoothness.

## Contribution

It provides a detailed analysis of Christoffel roots for nondetached $qP$ slowness surfaces, highlighting their elliptical shape and lack of smoothness compared to detached surfaces.

## Key findings

- Roots are elliptical but do not correspond to distinct wavefronts.
- $qP$ and $qSV$ slowness surfaces are not smooth.
- Detached surfaces have roots corresponding to distinct wavefronts.

## Abstract

The only restriction on the values of the elasticity parameters is the stability condition. Within this condition, we examine Christoffel equation for nondetached $qP$ slowness surfaces in transversely isotropic media. If the $qP$ slowness surface is detached, each root of the solubility condition corresponds to a distinct smooth wavefront. If the $qP$ slowness surface is nondetached, the roots are elliptical but do not correspond to distinct wavefronts; also, the $qP$ and $qSV$ slowness surfaces are not smooth.