Dispersive terms in the averaged Lagrangian method

In this paper we develop Whitham's formalism of the averaged Lagrangian in the problem of Stokes waves on the surface of a layer of ideal fluid by taking into account dispersive terms. We derive a general expression for the Lagrangian in which Whitham's term with the non-linear frequency of narrow-band wave trains is expressed in explicit form using derivatives of the complex amplitude phase of the first harmonic envelope. This Lagrangian form simplifies derivation of the evolution equations as variational equations.