Three-Dimensional Thermo-Acousto-Electric Diagonalized and Supplementary Equations in Fully Tri-Anisotropic and Inhomogeneous Media Part II: Relative Proof of Consistency

Abstract

The accompanying paper (Part I: Proof of Existence by Construction) in this transactions presents the diagonalization of coupled thermo-acousto-electric governing ( ${\mathcal {G}}$ ) and constitutive ( ${\mathcal {C}}$ ) systems of equations in fully tri-anisotropic and inhomogeneous media in three spatial dimensions. The construction of a set of associated supplementary partial differential equations has also been detailed in Part I. In this paper, distinct features of the diagonalized ( ${\mathcal {D}}$ ) and associated supplementary ( ${\mathcal {S}}$ ) systems of equations along with their constituent block operator matrices are scrutinized. It is rigorously shown that the constructed $\left({\mathcal {D}},{\mathcal {S}}\right)$ equations are internally consistent and sharply equivalent with the originating $\left({\mathcal {G}},{\mathcal {C}}\right)$ equations. Apart from firmly establishing the structural integrity of the $\left({\mathcal {D}},{\mathcal {S}}\right)$ equations, the provided (relative) proof of consistency illuminates the interwoven relationships between the $\left({\mathcal {D}},{\mathcal {S}}\right)$ and $\left({\mathcal {G}},{\mathcal {C}}\right)$ systems of equations.