Complementary fields conservation equation derived from the scalar wave equation

A conservation equation for the scalar wave equation is derived from two linearly independent solutions. In the one-dimensional limit the conservation equation yields a previously known invariant. The continuity equation derived for a complex disturbance is shown to yield an equivalent result. The obtention of the second independent solution is discussed using two different schemes that lead either to orthogonal trajectories or to derivative fields. The complementary fields may be visualized as out-of-phase fields where a negative-valued density is interpreted in terms of the leading or lagging field. These results are compared with the usual definition of energy density and flow for scalar waves. In the monochromatic plane wave case, the averages of all the proposed densities and flows converge to the same result. The physical meaning of the different approaches is discussed.