This section deals with the hydrodynamic fluid loading of a diffracting body in regular harmonic waves. The theory may be used to calculate the wave excitation on a fixed body or the wave exciting forces and radiation forces on a floating body.

Since the first order potential theory of diffraction and radiation waves is used here for radiation and diffraction analysis, the linear superposition theorem may be used to formulate the velocity potential within the fluid domain.

The fluid flow field surrounding a floating body by a velocity potential is defined by

(4–1)

where is the incident wave amplitude and is the wave frequency.

In Equation 4–1, the isolated space dependent term may be separated into contributions from the radiation waves due to six basic modes of body motion, the incident wave and the diffracted or scattered wave. The potential functions are complex but the resultant physical quantities such as fluid pressure and body motions in time domain analysis will be obtained by considering the real part only.

Adopting the conventional notation of the six rigid body motions in seakeeping theory, as demonstrated in Figure 1.4: Floating Rigid Motions , three translational and three rotational motions of the body's center of gravity are excited by an incident regular wave with unit amplitude:

(4–2)

The potential due to incident, diffraction, and radiation waves may therefore be written as:

(4–3)

where is the first order incident wave potential with unit wave amplitude, is the corresponding diffraction wave potential, is the radiation wave potential due to the j-th motion with unit motion amplitude.

In finite depth water, the linear incident wave potential at a point in Equation 4–3 has been given in Equation 2–2, but as a special case of unit amplitude .

When the wave velocity potentials are known, the first order hydrodynamic pressure distribution may be calculated by using the linearized Bernoulli's equation,

(4–4)

From the pressure distribution, the various fluid forces may be calculated by integrating the pressure over the wetted surface of the body. To have a general form for the forces and moments acting on the body, we extend the notation of unit normal vector of hull surface previously introduced through Equation 3–1 into 6 components corresponding to the six basic rigid body motions, such as

(4–5)

where is the position vector of a point on the hull surface with respect to the center of gravity in the fixed reference axes (FRA). Employing this notation, the first order hydrodynamic force and moment components can be expressed in a generalized form:

(4–6)

where is the mean wetted surface of body.

From Equation 4–3, the total first order hydrodynamic force can be written as

(4–7)

of which the j-th Froude-Krylov force due to incident wave is

(4–8)

the j-th diffracting force due to diffraction wave is

(4–9)

the j-th radiation force due to the radiation wave induced by the k-th unit amplitude body rigid motion is

(4–10)

Fluid forces can be further described in terms of reactive and active components. The active force, or the wave exciting force, is made up of the Froude-Krylov force and the diffraction force. The reactive force is the radiation force due to the radiation waves induced by body motions.

The radiation wave potential, , may be expressed in real and imaginary parts and substituted into Equation 4–10 to produce the added mass and wave damping coefficients

(4–11)

where the added mass and damping are

(4–12)

If a problem requires the wave loading on a fixed body, then only the active wave forces are of interest. When the body is floating, both the active and reactive fluid forces must be considered. It is also worth noting that all fluid forces calculated above are a function of the wetted body surface geometry only and are independent of the structural mass characteristics of the body.