Abstract
We 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.
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The authors wish to acknowledge Sandra Forte for fruitful discussions and David Dalton for insightful proofreading.
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This research was performed in the context of The Geomechanics Project supported by Husky Energy. Also, this research was partially supported by the Natural Sciences and Engineering Research Council of Canada, Grant 202259.
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Bos, L., Slawinski, M.A., Stanoev, T. et al. On orthogonal transformations of the Christoffel equations. Int J Geomath 11, 6 (2020). https://doi.org/10.1007/s13137-020-0141-7
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DOI: https://doi.org/10.1007/s13137-020-0141-7