= Solution
Substituting the sinusoidal <plane wave> into the <Navier-Cauchy equation> gives $\Delta u=-u$, $u_{tt}=-c^2u$, and $\nabla\operatorname{div}u=-k(k\cdot a)\sin(k\cdot X-ct)$ because $|k|=1$. Equality of the amplitudes is therefore
$$
\boxed{\rho c^2a=\mu a+(\lambda+\mu)k(k\cdot a).}
$$
This is an <eigenvalue problem> for the <acoustic tensor> $\mu I+(\lambda+\mu)k\otimes k$. Choosing a nonzero <amplitude> parallel to $k$ gives $\rho c^2=\lambda+2\mu$; choosing any nonzero <amplitude> perpendicular to $k$ gives $\rho c^2=\mu$. For positive <mass density> and stable <Lamé parameters>, both squared speeds are positive. These choices construct nonzero solutions explicitly, proving existence.
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