Abstract The W. K. B. approximation is applied to a general system of linear partial differential equations which may be derived from a variational principle of a certain type. The theory describes slowly varying wavetrains, with the oscillation locally in one of the normal modes of a waveguide of quite general structure. The governing equations need not be hyperbolic; the wavelike character of the solution may be imparted by the lateral boundary conditions in the waveguide (e. g. surface waves on water). Variations in amplitude of the waves along rays are governed by conservation of an adiabatic invariant, as suggested by Whitham’s averaged variational principle. Higher order approximations may be constructed and the equations integrated b y quadrature. The averaged variational principle is also derived directly, in a manner applicable also to general nonlinear systems. It is shown to be a necessary condition governing the lowest order approximation for an asymptotic expansion of the same type as that for linear systems, provided such an expansion exists. However, it is not clear from this second approach how to construct higher order approximations.