Using a reference frame moving with the forward speed of a structure, the coordinate of a point in this reference frame satisfies

(4–37)

where is the forward speed of structure with respect to the fixed reference axes (FRA), is the coordinate of the point with respect to the FRA, and is the coordinate of the point with respect to the moving reference frame.

The total unsteady fluid potential varies with the encounter frequency:

(4–38)

where the encounter frequency can be given as

(4–39)

where is the incident wave number, , and is the heading angle between the vessel forward speed and the wave propagation direction.

In this moving reference frame, if the disturbed steady flow is neglected, the linear free surface equation is satisfied, such that

(4–40)

and the body surface conditions

(4–41)

are satisfied on the mean wetted body surface , where

(4–42)

Based on the Bernoulli equation, the first order hydrodynamic pressure is:

(4–43)

Similar to Equation 4–7 through Equation 4–11, we have the j-th Froude-Krylov force due to incident wave

(4–44)

the j-th diffracting force due to diffraction wave

(4–45)

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

(4–46)

Note that the frequency domain pulsating Green's function source distribution method, as described in Source Distribution Method, does not account for forward speed in its formulation. The translating-pulsating source method, contrary to the pulsating source method, explicitly accounts for forward speed in its formulation and should be used for the hydrodynamic analysis for cases where forward speed is simulated. The numerical evaluation of the translating-pulsating Green's function is very time-consuming. When the forward speed is small in amplitude, an approximate free surface boundary condition can be used, for example

(4–47)

Under this situation, the frequency domain pulsating Green's function can be employed together with the new body boundary condition given in Equation 4–41 and Equation 4–42 to numerically solve the diffraction and radiation potential components. The wave exciting forces, added mass, and damping can be estimated from Equation 4–44 , Equation 4–45, Equation 4–46 afterwards.

This approximate pulsating source method has been extensively tested against the translating-pulsating Green's function method. It was found that, although the translating-pulsating source gives benefits in the calculation of individual hydrodynamic coefficients and wave action, the differences in the response calculations are quite small in cases where low to moderate speeds are considered (i.e. ). It should be noted that the computational effort required for the translating-pulsating source far exceeds that for the pulsating source methods (Inglis & Price [19]).

This approximate method can also be applied for hydrodynamic analysis of multiple structures travelling side by side with same constant forward speed. Similar to Equation 4–35, the wave exciting force and the added mass and damping coefficients are written as

(4–48)