= Solution
Assume a stable, lossless elastic body with the usual symmetry of the <elasticity tensor> and <boundary conditions> that eliminate the surface terms in <integration by parts>, such as a traction-free outer surface. Its elastic stiffness operator is then self-adjoint. When <density> varies spatially, the precise eigenproblem is generalized:
$$
\hat H\mathbf u_\alpha=\rho\omega_\alpha^2\mathbf u_\alpha,
\qquad
\langle\mathbf u_\alpha,\mathbf u_\beta\rangle_\rho
=\int_\Omega\rho\,\mathbf u_\alpha^*\cdot\mathbf u_\beta\,dV
=\delta_{\alpha\beta}.
$$
Here $\alpha$ abbreviates the pair of mode labels. Equivalently, $\rho^{-1}\hat H$ is a <self-adjoint operator> in the mass <inner product>. Stability makes $\omega_\alpha^2$ real and nonnegative. Modes at different <eigenvalues> are orthogonal, and an <orthonormal basis> can be selected within each degenerate eigenspace. Completeness permits expansion of a suitable <displacement field> in these <mass-weighted elastic normal modes>.
Insert the expansion and project onto $\mathbf u_\alpha^*$. Mass normalization diagonalizes both the <acceleration> term and the restoring term. For a symmetric <stress glut tensor> supported away from the boundary,
$$
\int_\Omega\mathbf u_\alpha^*\cdot(-\nabla\cdot\boldsymbol\Gamma)\,dV
=\int_\Omega\boldsymbol\Gamma:\nabla\mathbf u_\alpha^*\,dV
=\int_\Omega\boldsymbol\Gamma:\mathbf e_\alpha^*\,dV.
$$
The symmetry of the <stress glut tensor> removes the antisymmetric part of the displacement gradient, leaving the <strain>. The result is \b[one independent forced oscillator per mode], with the stated excitation integral. If the modes instead have mass norm $N_\alpha$, that integral is divided by $N_\alpha$. A source touching the boundary requires the corresponding surface forcing to be retained rather than silently dropping it.
Rigid translations and rotations can be zero-frequency modes of a free body. Their <strain> vanishes, so an internal symmetric <seismic moment tensor> does not excite them through this coupling. Attenuation, radiative boundaries or other non-self-adjoint effects require modifications of the simple conservative expansion.
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