= Solution
Resolve each <amplitude> by the <orthogonal projections>
$$
a_L=(a\cdot k)k,\qquad a_T=a-(a\cdot k)k.
$$
The <acoustic tensor> acts by $\lambda+2\mu$ on the longitudinal line and by $\mu$ on its perpendicular plane. Thus
$$
\boxed{c_L=\sqrt{\frac{\lambda+2\mu}{\rho}},\qquad c_T=\sqrt{\frac\mu\rho}.}
$$
The longitudinal component is a <longitudinal wave>, while the two independent transverse polarizations are <transverse waves>. Each projection obeys its own <wave equation>, and <superposition> adds the corresponding solutions.
A general planar disturbance is therefore a sum of the two types. For a single sinusoid with a common speed, both projections cannot be nonzero unless the two speeds coincide: projecting the <eigenvalue equation> gives $(\rho c^2-\lambda-2\mu)a_L=0$ and $(\rho c^2-\mu)a_T=0$. This distinguishes <superposition> of <waves> with different speeds from a mixed polarization traveling rigidly at one speed.
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