Plane waves of spin angular momentum density in an ideal elastic solid are analyzed using vector and bispinor descriptions. In both classical and quantum physics, spin density is the axial vector field whose curl is equal to twice the incompressible intrinsic momentum density. The second-order vector wave equation assumes that temporal changes of spin density in an ideal elastic solid are attributable to convection, rotation, and torque density. The corresponding first-order wave equation for Dirac bispinors incorporates terms describing wave propagation, convection, rotations of the medium and rotations of wave velocity relative to the medium. The two rotation terms are also operators for rotational kinetic energy and conventional potential energy, respectively. The potential energy corresponds to half the mass term of the free electron Dirac equation. Bispinor plane wave solutions are constructed consistent with the usual dynamical operators of relativistic quantum mechanics. Lagrangian and Hamiltonian densities are also constructed with each term having a clear classical physics interpretation. The intrinsic momentum associated with the Belinfante-Rosenfeld stress tensor is explained. Application to elementary particles is discussed, including classical physics analogues of the Pauli exclusion principle, interaction potentials, fermions, bosons, and antimatter.