= Solution
Let $\eta=\hat{\mathbf k}\cdot\mathbf x-ct$. Then
$$
\nabla^2\mathbf u=\mathbf f''(\eta),
\qquad
\nabla(\nabla\cdot\mathbf u)
=\hat{\mathbf k}(\hat{\mathbf k}\cdot\mathbf f''(\eta)),
$$
and substitution gives
$$
\rho c^2\mathbf f''
=(\lambda+\mu)\hat{\mathbf k}
(\hat{\mathbf k}\cdot\mathbf f'')+\mu\mathbf f''.
$$
For a P-wave, $c=c_P$ and the varying part of $\mathbf f$ is longitudinal:
$$
\mathbf f(\eta)=\hat{\mathbf k}F(\eta),
\qquad
\theta=F'(\eta),
\qquad
\boldsymbol\omega=0.
$$
For an S-wave, $c=c_S$ and $\mathbf f$ is transverse:
$$
\boxed{\hat{\mathbf k}\cdot\mathbf f=0,
\qquad
\theta=0,
\qquad
\boldsymbol\omega=\hat{\mathbf k}\times\mathbf f'(\eta).}
$$
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