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Distinct squared traction-free eigenfrequencies in a bounded lossless elastic body have eigenfunctions orthogonal in $L^2(\mathcal{D},\rho dV{)}^3$, by the Betti identity for elastodynamic fields. Normalize one field from each eigenspace. This is an orthonormal set in a separable Hilbert space, so it is countable: disjoint balls of radius less than $1/\sqrt{2}$ around its unit vectors must contain distinct points of a countable dense set. Positive and negative frequency partners add at most a factor of two. This argument proves countability without requiring a separate compact-resolvent proof.

Use the same real elastic stiffness tensor for both fields, including its minor and major symmetries, and let $t_i^a={\hat{\sigma }}_{ij}^a{n}_j$, with outward unit boundary normal $n_j$. No complex conjugation is needed for the initial bilinear identity. DefineThe divergence has two force-balance terms and two cross-gradient terms. The latter cancel becausethis follows by exchanging the two strain-gradient factors using major symmetry and the minor pair symmetries. The frequency-domain equations then giveIntegrating and applying the divergence theorem provesThis is the Betti identity for elastodynamic fields. Spatially varying stiffness is allowed because its derivatives remain inside the divergence. The original PDF's constitutive contraction is $c_{ijpq}{\partial }_p{\hat{u}}_q$; the TeX's repeated displacement field index is erroneous.