  This paper provides an alternative methodology for analysis of three-wave
interactions under the exact dispersion relation associated with gravity waves
in fluid of intermediate depth. A Korteweg-de Vries type of equation with exact
dispersion property is adopted as the governing equation for unidirectional
wave packet evolution. Following the idea from Zakharov's seminal paper
(Zakharov, V. E. (1968) Stability of periodic waves of finite amplitude on the
surface of a deep fluid. \textit{Journal of Applied Mechanics and Technical
Physics}, {\bf 9}, 190--194), the equation is transformed from the
spatial--temporal domain to the wavenumber--temporal domain. The solution of
the transformed equation is sought using the perturbation theory, for which the
ansatz is expressed in the form of a regular expansion in the increasing order
of a small parameter. After implementing the na\"{i}ve perturbation method, due
to nonlinear mode generation and particular combinations of wavenumbers, the
third-order solution contains spurious secular growth terms which appear as a
consequence of resonant interaction and nonlinear mode generation. These
spurious secular growth terms can be prevented by implementing the method of
strained parameters for which nonlinear dispersion relation terms are produced
for particular combination of wavenumbers.
