# On orthogonal transformations of Christoffel equations

## TL;DR

This paper proves the equivalence of different forms of Christoffel equations under orthogonal transformations, broadening the understanding of wave propagation in elastic solids beyond traditional assumptions.

## Contribution

It introduces a general proposition that establishes the invariance of Christoffel equations under orthogonal transformations, without restrictions on space dimension or tensor symmetries.

## Key findings

- Proves equivalence of Christoffel equations under rotations and orthogonal transformations.
- Extends validity beyond R^3 and symmetric elasticity tensors.
- Provides a general mathematical framework for wave propagation analysis.

## Abstract

The purpose of this paper is to prove the equivalence$-$under rotations of distinct terms$-$of different forms of a determinantal equation that appears in the studies of wave propagation in Hookean solids, in the context of the Christoffel equations. To do so, we prove a general proposition that is not limited to ${\mathbb R}^3$, nor is it limited to the elasticity tensor with its index symmetries. Furthermore, the proposition is valid for orthogonal transformations, not only for rotations. The sought equivalence is a corollary of that proposition.