TABLE OF CONTENTS The evolution of an initially two-dimensional, long-crested wave of dimensional finite wave height 
and dimensional wave period 
in a fluid of constant dimensional finite-depth 
may be analyzed by the method of multiple scales (MMS) (Nayfeh, 1981). Applications of the MMS to surface gravity waves are given by Chu and Mei (1970 and 1971), Davey (1972) and Whitham (1974) (vide., Mei, 1989 for an extensive bibliography). The MMS has also been applied with much success to the Lagrangian variational approach of Luke (1967). An application of MMS to surface gravity waves over wavy bottoms is given by Kirby (1986 a, b); and applications of MMS to cross-waves in both rectangular and circular wave basins are given by Miles and his colleagues (Becker and Miles, 1992 and Miles and Becker, 1988).
The dimensional fluid velocity 
in a fixed coordinate system 
may be be computed as in Sec. 3 from the negative gradient vector of a dimensional scalar velocity potential 
by (vide., Chapter 2.2.7)
The scaling of the boundary-value-problem BVP is the same as in Sec. 3 with the exception that the dimensional amplitude 
scale will be replaced by the dimensional wave height 
/2 scale so that the perturbation ordering parameter is now 
. In addition, the Cartesian coordinate system of Sec. 3 will now be a fixed dimensional inertial coordinate system with the 
and 
axes horizontal at the still water level (SWL) with a...
