  Stokes perturbative solution of the nonlinear (boundary value dependent)
surface gravity wave problem is known to provide results of reasonable accuracy
to engineers in estimating the phase speed and amplitudes of such nonlinear
waves. The weakling in this structure though is the presence of aperiodic
secular variation in the solution that does not agree with the known periodic
propagation of surface waves. This has historically necessitated increasingly
higher ordered (perturbative) approximations in the representation of the
velocity profile. The present article ameliorates this long standing
theoretical insufficiency by invoking a compact exact $n$-ordered solution in
the asymptotic infinite depth limit, primarily based on a representation
structured around the third ordered perturbative solution, that leads to a
seamless extension to higher order (e.g. fifth order) forms existing in the
literature. The result from this study is expected to improve phenomenological
engineering estimates, now that any desired higher ordered expansion may be
compacted within the same representation, but without any aperiodicity in the
spectral pattern of the wave guides.
