Conventional descriptions of transverse waves in an elastic solid are limited by an assumption of infinitesimally small gradients of rotation. By assuming a linear response to variations in orientation, we derive an exact description of a restricted class of rotational waves in an ideal isotropic elastic solid. The result is a nonlinear equation expressed in terms of Dirac bispinors. This result provides a simple classical interpretation of relativistic quantum mechanical dynamics.We construct a Lagrangian of the form $${\fancyscript{L} = -\fancyscript{E} + U + K = 0}$$ , where $${\fancyscript{E}}$$ is the total energy, U is the potential energy, and K is the kinetic energy.