= Betti identity for elastodynamic fields
{c}
{title2=$\nabla\cdot(\boldsymbol\sigma^a\mathbf u^b-\boldsymbol\sigma^b\mathbf u^a)=\rho(\omega_b^2-\omega_a^2)\mathbf u^a\cdot\mathbf u^b+\mathbf u^a\cdot\mathbf f^b-\mathbf f^a\cdot\mathbf u^b$}
For two Fourier <elastic wave> fields with the same real <elastic stiffness tensor>, its major symmetry cancels the cross-strain terms in the <divergence> displayed above. The force-balance equations supply the remaining volume terms, and the <divergence theorem> gives their boundary integral of reciprocal <tractions>. Taking one field to be a <complex conjugate> gives energy-work identities and weighted modal <orthogonality>. The bilinear identity uses no <complex conjugation> until that field is chosen explicitly.
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