= Solution
A small <displacement> $\mathbf u$ in a homogeneous isotropic elastic solid obeys <linear elasticity>. With Lamé constants $\lambda,\mu$ and <mass> <density> $\rho$, <momentum> balance gives
$$
\rho\partial_t^2\mathbf u=(\lambda+\mu)\nabla(\nabla\cdot\mathbf u)+\mu\nabla^2\mathbf u.
$$
A longitudinal <plane wave> has <displacement> parallel to its propagation direction and changes local volume. A <transverse wave> has perpendicular <displacement> and produces shear. Substitution gives
$$
\boxed{c_L=\sqrt{\frac{\lambda+2\mu}{\rho}},\qquad
c_T=\sqrt{\frac\mu\rho}.}
$$
An ideal fluid has no static <shear modulus> and cannot support this bulk transverse <elastic wave>. A thin rod with free lateral surfaces instead supports an extensional long-wavelength mode with speed $\sqrt{E/\rho}$; its transverse contraction changes the effective stiffness, so it should not be identified with the unbounded-solid longitudinal mode.
<Stress> and <displacement> boundary conditions produce reflection, transmission, mode conversion and guided <waves>. <Viscoelasticity> and scattering can attenuate <waves>; finite geometry can make guided modes dispersive even when the bulk elastic constants are frequency independent. \b[<Wave> speed depends on both material stiffness and the allowed deformation.]
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