 Mass-weighted elastic normal modes

For a conservative elastic body, the stiffness operator is a self-adjoint operator under compatible boundary conditions. The dynamical operator ${\rho }^{-1}\hat{H}$ is self-adjoint in the mass inner product $⟨\mathbf{u},\mathbf{v}{⟩}_{\rho }=\int \rho {\mathbf{u}}^{\ast }\cdot \mathbf{v}dV$. Choosing mass-normalized normal modes converts the field equation into independent forced oscillators. If the normal modes have mass norm $N_{\alpha }$ instead, each projected force must be divided by $N_{\alpha }$.