  Using the wave equation as an example, it is shown how to extend the
hydrodynamic Lagrangian-picture method of constructing field evolution using a
continuum of trajectories to second-order theories. The wave equation is
represented through Eulerian-picture models that are distinguished by their
Lorentz transformation properties. Introducing the idea of the relativity of
the particle label, it is demonstrated how the corresponding trajectory models
are compatible with the relativity principle. It is also shown how the Eulerian
variational formulation may be obtained by canonical transformation from the
Lagrangian picture, and how symmetries in the Lagrangian picture may be used to
generate Eulerian conserved charges.
