Nonlinear hydrostatic and Froude-Krylov forces, as well as Morison forces, are evaluated by integrating the fluid pressure over the instantaneous wetted surface or by employing the Morison equation on tube/disc elements of a floating structure in rough waves. Accurate dynamic or kinematic properties of fluid particles beneath the wave surface are thus required for this purpose.
Du et al. [13] discussed and developed several approaches for a more accurate estimation of the fluid pressure and fluid particle velocity. Among these approaches, Wheeler stretching [43] for the first order approximation, as well as extended Wheeler stretching for the second order approximation, are selected for their efficiency and ease of completion in an Aqwa time domain analysis of severe waves. These ensure that the total pressure at the instantaneous incident wave surface (consisting of the incident wave pressure and the hydrostatic pressure) is zero.
This first order wave formula for an incident wave is derived from the perturbation expression beneath the mean free surface. The main properties of a regular linear Airy wave in finite-depth water, such as the fluid potential, hydrodynamic pressure, velocity and acceleration of a fluid particle, and the wave elevation, are expressed in first order terms:
 | (13–37) |
where 
is the wave amplitude, 
is the wave frequency (in rad/s), 
is the wave number, 
is the wave propagation
direction, 
is the wave phase, 
is the water depth, and 
is the position vector in the
fixed reference axes.
The linear perturbation expression beneath the mean free surface
implies that a point defined by 
in Equation 13–37,
at a given time 
, is not exactly at that physical location but resides
at the equilibrium position where the velocity potential is determined.
The exact location corresponding to this equilibrium position 
at time 
is thus denoted as 
in the fixed reference axes, of which the
origin is on the mean water level, and where the z-axis points upwards.
Employing Wheeler stretching [43],
the coordinate transformation from 
to 
is
 | (13–38) |
Rotate the coordinate system about the z-axis such that the x-axis points in the wave propagation direction:
 | (13–39) |
With these transformations, the main properties of a regular
linear Airy wave at a specified location 
beneath
the instantaneous incident wave surface are simplified as
 | (13–40) |
where 
and 
is the sum of the incident wave pressure and the
hydrostatic pressure.
For a deep water case where 
, the Wheeler stretching transformation
is simplified as
 | (13–41) |
The main properties of a regular linear Airy wave in deep water are then expressed as
 | (13–42) |
From Equation 13–40 and Equation 13–42, it is
easily proved that 
at a point 
on the instantaneous incident wave surface, for both the finite-depth and
deep water cases.
Employing Equation 13–39, irregular linear long-crested waves can be expressed as the summation of a series of regular incident waves based on the first order perturbation beneath the mean free surface, that is,
 | (13–43) |
where 
is the total number of wave components.
Employing Wheeler stretching (Equation 13–38) at an actual location 
in the
fixed reference axes (FRA), at time 
, Equation 13–43 can be extended for irregular long-crested waves in finite-depth
water, such as
 | (13–44) |
where 
.
For deep water irregular waves, the Wheeler stretching given by Equation 13–41 is used, such that Equation 13–44 may be simplified as
 | (13–45) |
The potential and wave elevation of second order Stokes waves with set-down terms are given in Equation 2–15.
The velocity of a fluid particle at the equilibrium position 
is
 | (13–46) |
The acceleration of a fluid particle up to the second order may be written as:
 | (13–47) |
where 
is assumed to be a small
perturbation variable.
Employing Wheeler stretching, the real parts of the fluid pressure components up to the second
order at a point 
are expressed as:
 | (13–48) |
Note that in Equation 13–48, a new partial differential term
is introduced to account for the first order dynamic pressure variation in
the vertical direction, from the equilibrium position to the actual position.
The explicit forms of the real parts of the fluid particle velocity, acceleration and pressure terms are
 | (13–49) |
With the above form, the real part of the fluid particle velocity
at time 
is
 | (13–50) |
While the real part of the acceleration is
 | (13–51) |
The first order part of 
is
 | (13–52) |
Including only the first order term, the real part of the extended hydrodynamic pressure is
 | (13–53) |
From Equation 13–53, it is observed that the hydrodynamic incident wave pressure not only includes the partial derivatives of the first and second order potentials with respect to time, but also has the nonlinear Bernoulli term and a new perturbation term against the vertical shifting of the location. This equation is employed directly to calculate the nonlinear Froude-Krylov and hydrostatic forces on a floating structure modeled by a series of panels.
For the deep water case, where 
, the velocity and acceleration of a fluid
particle at the equilibrium position 
is
 | (13–54) |
The real parts of the modified hydrodynamic pressure and the hydrostatic pressure beneath the instantaneous incident wave surface are
 | (13–55) |