In a time domain analysis, the wave directional angles and in Equation 5–20 should be treated as the relative directions between the wave propagating direction and the vessel orientation at each time step. Therefore the frequency domain database of covering all the possible relative directions should be created prior to any time domain analysis. The values of at any actual vessel position at a time step could be then estimated by the means of database interpolation. However even with this database interpolation treatment, the quadruple summation form given in Equation 5–20 is still prohibitively difficult to apply for the numerical time domain simulation procedure due to the large processing and memory requirements.

For the unidirectional wave case, Newman's approximation [28] is frequently used in practice. This takes the off-diagonal difference frequency QTF value to be an average of the corresponding diagonal values:

(5–24)

The out of phase item is excluded in this approximation for the difference frequency second order force calculation as .

Newman's approximation is normally accepted for the hydrodynamic analysis of moored offshore structures in moderate and deep water depth in long crested waves.

In Aqwa, Newman's approximation is extended to the multiple directional wave case for the difference frequency QTF element evaluation:

(5–25)

In the above equation, the out of phase item is also defined, because are not necessarily zero. However it is easily observed that this extended approximation will be exactly the same as the original one shown in Equation 5–24 when long crested waves are concerned (). With the simplified expressions of the difference frequency QTF elements in Equation 5–25, the symmetric properties shown in Equation 5–19 still remain true with respect to a pair of waves with difference frequencies and wave directions:

(5–26)

Denoting and and substituting Equation 5–25 into Equation 5–20 for the difference frequency second order force and moment only, we have

(5–27)

Comparing the equation above to Equation 5–20 it is observed that the summation against the frequencies of the m-th directional waves has been uncoupled from summations against the frequencies of the n-th (n = 1, N_d ) directional waves, which converts the quadruple summations into triple summations and hence greatly increases the numerical calculation efficiency. In addition, instead of using the four dimensional elements of for obtaining the difference frequency drift force, the directional coupling mean QTF matrices of are required when employing Equation 5–27, which significantly reduces the memory buffer and hard disk requirements.