  In this paper we introduce a new method for exact decomposition of
propagating, nonlinear magnetohydrodynamic (MHD) disturbances into their
component eigenenergies associated with the familiar slow, Alfv\'en, and fast
wave eigenmodes, and the entropy and field-divergence pseudo-eigenmodes. First
the mathematical formalism is introduced, where it is illustrated how the
ideal-MHD eigensystem can be used to construct a decomposition of the time
variation of the total energy density into contributions from the eigenmodes.
The decomposition method is then demonstrated by applying it to the output of
three separate nonlinear MHD simulations. The analysis of the simulations
confirms that the component wave modes of a composite wavefield are uniquely
identified by the method. The slow, Alfv\'en, and fast energy densities are
shown to evolve in exactly the way expected from comparison with known linear
solutions and nonlinear properties, including processes such as mode
conversion. Along the way, some potential pitfalls for the numerical
implementation of the decomposition method are identified and discussed. We
conclude that the exact, nonlinear decomposition method introduced is a
powerful and promising tool for understanding the nature of the decomposition
of MHD waves as well as analysing and interpreting the output of dynamic MHD
simulations.
