# On commutativity and near commutativity of translational and rotational averages: Analytical proofs and numerical examinations

## TL;DR

This paper analytically and numerically investigates the conditions under which translational and rotational averages of elasticity parameters commute, revealing near commutativity in weakly anisotropic materials relevant to seismology.

## Contribution

It provides analytical proofs and numerical analysis showing when and how translational and rotational averages commute or nearly commute based on symmetry and anisotropy strength.

## Key findings

- Averages do not generally commute, except in special symmetry cases.
- Noncommutativity depends on the degree of anisotropy.
- Weak anisotropy leads to near commutativity, with the commutator proportional to the square of anisotropy strength.

## Abstract

We show that, in general, the translational average over a spatial variable---discussed by Backus \cite{backus}, and referred to as the equivalent-medium average---and the rotational average over a symmetry group at a point---discussed by Gazis et al. \cite{gazis}, and referred to as the effective-medium average---do not commute. However, they do commute in special cases of particular symmetry classes, which correspond to special relations among the elasticity parameters. We also show that this noncommutativity is a function of the strength of anisotropy. Surprisingly, a perturbation of the elasticity parameters about a point of weak anisotropy results in the commutator of the two types of averaging being of the order of the {\it square\/} of this perturbation. Thus, these averages nearly commute in the case of weak anisotropy, which is of interest in such disciplines as quantitative seismology, where the weak-anisotropy assumption results in empirically adequate models.