Betti identity for elastodynamic fields ($\nabla \cdot ({\mathbf{\sigma }}^a{\mathbf{u}}^b-{\mathbf{\sigma }}^b{\mathbf{u}}^a)=\rho ({\omega }_b^2-{\omega }_a^2){\mathbf{u}}^a\cdot {\mathbf{u}}^b+{\mathbf{u}}^a\cdot {\mathbf{f}}^b-{\mathbf{f}}^a\cdot {\mathbf{u}}^b$)

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.