In a moving reference frame with the same forward speed as the structure, the fluid particle velocity is expressed as

(4–53)

where the unsteady fluid potential is defined in Equation 4–37 and Equation 4–38 and includes the incident, diffraction and radiation potential components.

Based on the first order wave potential, the fluid pressure (up to the second order) at a position on the instantaneous wave surface is

(4–54)

Taking the perturbation to the second order with respect to the corresponding location on the mean wave surface, , we have

(4–55)

Note that the second order term of is ignored to avoid calculating the second order derivatives of the unsteady wave potential.

From Equation 4–38, it is found that

(4–56)

as the fluid pressure on the actual disturbed wave surface must be zero, from Equation 4–55 and Equation 4–56 the disturbed wave elevation with the second order correction can be derived as

(4–57)

If only the first order disturbed wave elevation is required, Equation 4–57 is simplified as

(4–58)

If only the linear incident wave elevation is required, Equation 4–58 is further simplified as

(4–59)

Equation 4–57, Equation 4–58, and Equation 4–59 are applicable for both single structure and multi-body hydrodynamic interaction cases with or without forward speed. For a case without forward speed, .

From Equation 4–51 and Equation 4–57, the air gap at a point P of on the m-th structure is:

(4–60)