where *G*^{S} represents a local flow disturbance defined in terms of simple (Rankine) singularities as

4*πG*^{S}=–1/*r*+1/*r*′ (3)

with

and *G*^{F} accounts for free-surface effects. The free-surface component *G*^{F} is given by

(4)

with

Here, and respectively stand for the singularity and flow-observation points. Furthermore, *D*_{ε} is the *dispersion function*

*D*_{ε}*=(f+iε–Fα)*^{2}–*k* (5a)

where is the wavenumber. The function *D*_{ε} is equal to the real dispersion function *D*

*D=(f–Fα)*^{2}–*k* (5b)

in the limit *ε*=0.

The velocity potential at a point of the mean wetted surface of the ship (where the normal derivative *∂**/∂n* is presumed known) is defined by the solution of an integral equation of the form

/2+*χ*=*ψ* (6a)

where the potentials *ψ* and *χ* respectively correspond to a-priori given and unknown distributions of sources and normal dipoles, with densities determined by the values of *∂**/∂n* and *,* at the mean wetted hull *H* and the mean waterline *W* of the ship. The potential at a point in the mean flow domain outside *H* is given by

*=ψ–χ.* (6b)

The source and dipole potentials *ψ* and *χ* in the integral equation (6a) and the flow representation (6b) can be expressed as

(7)

where the potentials *ψ*^{S} and *χ*^{S} correspond to the simple-singularity component *G*^{S} in expression (2) for the Green function, and the components *ψ*^{F} and correspond to the free-surface component *G*^{F}. Thus, the components *ψ*^{S} and *χ*^{S} and the components *ψ*^{F} and defined by (16) and (17–18) in [3], are expressed in terms of source and dipole distributions involving the simple singularity *G*^{S} and the free-surface term *G*^{F}*,* respectively.

The free-surface potentials *ψ*^{F} and in (7) are now considered. In the usual *free-surface Green-function approach,* based on the Green function (2) associated with the free-surface condition (1), the Green function is evaluated and subsequently integrated over the ship hull and waterline to determine the source and dipole potentials *ψ* and *χ*. The well-known difficulties (stemming from the complex singularity of the free-surface component *G*^{F} for *x=ξ,y=η,z+ζ=0*) involved in this classical approach are partially circumvented in Kochin's formulation, in which the surface and line distributions of sources and dipoles over the ship hull and waterline defining the free-surface components *ψ*^{F} and are considered directly.

Within the Fourier-Kochin formulation, the potentials and are defined by (24) and (12) in [3] as

(8)

where *N* and are spectrum functions defined by distributions of the elementary wave function *ε* over the mean ship hull *H* and waterline *W.* The spectrum function associated with the potential *χ* in the integral equation (6a), is called the *kernel spectrum function*.

Two alternative mathematical representations, called the *potential representation* and the *velocity representation* because they respectively involve the potential and the velocity ∇*,* of the kernel spectrum function are given in [3]. The potential representation follows from Kochin's formulation in a straightforward way. The velocity representation is obtained in [3] from the potential representation via an integration by parts based on Stokes' theorem. The velocity representation of the kernel spectrum function is shown in [3] to provide a substantially more solid mathematical basis than the potential representation for the purpose of numerically evaluating influence coefficients. The velocity representation is also preferable for coupling viscous and potential flows, as is shown further on. Thus, only the velocity representation is considered here.

The velocity representation of the spectrum function is defined by (65), with Δ=0, in [3]. A useful extension of this representation is obtained