By doing the perturbation up to the second order at a location close to the mean wave surface, the wave elevation and its partial derivatives can be further written as

(13–56)

where is the first order incident wave elevation, is the first order incident wave fluid particle velocity at the mean water surface, and is the first order incident wave fluid particle acceleration at the mean water surface.

Denoting as a field point beneath the instantaneous incident wave surface in the fixed reference axes, the Wheeler stretching at this location is determined by

(13–57)

Where is the water depth and the wave elevation up to the second order is given by Equation 13–56.

The second order correction of the dynamic pressure at this point is written as:

(13–58)

Similarly, the second order correction of the fluid particle velocity and acceleration are expressed as

(13–59)