This section is only concerned with the second order force and moment due to the first order wave potential and the first order motion responses. The second order fluid potential components shown in Equation 5–6 will be discussed separately in Second Order Wave Potential and Its Simplification.
Extending the unidirectional incident wave definition to a more
general multiple directional wave case, the fluid wave potential
of an incident wave in the m-th
wave direction is expressed in the complex form. For example, the
first order incident regular wave elevation at a point on the mean
water surface 
is represented
as
 | (5–13) |
where 
is the wave amplitude, 
is the frequency, 
is the wave
number, 
is the direction, and 
is the phase.
All the variables in the second order force and moment expressions in Equation 5–6 are real numbers. The complex forms of the first order relative wave elevation, potential, displacement and acceleration of the body corresponding to the unit wave amplitude are:
 | (5–14) |
The second order wave exciting force/moment due to the first
order waves and motion responses resulting from a pair of the regular
incident waves with (
, 
, 
, 
) and (
, 
, 
, 
) can be written as
 | (5–15) |
where the coefficients for the second order wave force are:
 | (5–16) |
and the coefficients for the second order wave moment are:
 | (5–17) |
From Equation 5–15, 
are used as the sum frequency
force components, while 
contribute only in
the difference frequency force components. In cases with multiple
directional irregular waves, the total second order wave exciting
force (not including the component due to the second order potential)
has a quadruple summation form,
 | (5–18) |
where 
is the number of wave directions, and 
and 
are the numbers of wave components in the m-th and the n-th wave
directions respectively.
Further introducing a new set of matrices for a generalized multi-directional wave set, in which the so-called composite elements are defined:
 | (5–19) |
in which the composite elements 
are symmetric against a pair of the waves with (
, 
, 
, 
) and (
, 
, 
, 
) and 
is skew-symmetric against this pair of waves.
Employing above definitions, Equation 5–18 can be rewritten as
 | (5–20) |