= Solution
For a uniform material and an affine static base deformation, the <deformation gradient>, <incremental elastic moduli> and reference <mass density> are constant. With zero incremental <body force>, the equation reduces to $c_{\alpha i\beta j}v_{j,\beta\alpha}=\rho_0\ddot v_i$. Let $\nu$ be a unit material propagation direction, $m$ a constant polarization and $s=t-\nu_\alpha\xi_\alpha/c$. Direct differentiation of the <plane wave> gives
$$
v_i=m_if(s),\qquad \ddot v_i=m_if''(s),\qquad v_{j,\beta\alpha}=\frac{\nu_\alpha\nu_\beta}{c^2}m_jf''(s).
$$
Thus every twice differentiable profile $f$ satisfies the incremental equation if
$$
\boxed{Q_{ij}(\nu)m_j=\rho_0c^2m_i,\qquad Q_{ij}(\nu)=c_{\alpha i\beta j}\nu_\alpha\nu_\beta.}
$$
This is the <acoustic tensor> <eigenvalue> problem. The <acoustic tensor> is a real <symmetric matrix>: interchange the index pairs in the <incremental elastic moduli> and then interchange the dummy material indices to obtain $Q_{ji}=Q_{ij}$. Its <eigenvectors> give the displacement polarizations. A positive <eigenvalue> gives a real nonzero speed $c$; arbitrary-profile propagation imposes precisely this algebraic condition. Boundary conditions determine which combinations of these local <plane waves> are allowed in a finite body.
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