∇_{ξ}^{F}*, ψ*^{F} and *χ*^{F} in the decompositions (16) and (12) into simple-singularity and free-surface components. These components are examined in [4,5] and in this study. Specifically, the Fourier representation of free-surface effects defined by (20) is considered for a generic spectrum function *S* and a generic dispersion function *D.* The generic Fourier integral (20) is expressed in (21) as

*φ*^{F}*=φ*^{W}*+φ*^{N}

where *φ*^{W} and *φ*^{N} respectively correspond to a wave component and a near-field component. The wave component *φ*^{W} is given by (26), a single Fourier integral along the *dispersion curve(s)* defined in the Fourier plane by the dispersion relation *D*=0. The Fourier representation (26) is a generalization of the representation given in [4], which is obtained if *E*=1 and (24a) is used for Σ2. The near-field component *φ*^{N} in the decomposition (21) is significant in the near field but is negligible in comparison to the wave component *φ*^{W} in the far field. The component *φ*^{N} corresponding to the wave component (26) is defined by (27), which is a generalization of (30) in [5]. This expression for the near-field component *φ*^{N} is shown in [5] to be well suited for accurate numerical evaluation.

The Fourier representation of (20) defined by (21), (26) and (27) is valid for generic dispersive waves and for arbitrary singularity distributions, including the special case of the Green function (i.e. a point source). Applications of this generic Fourier representation to wave diffraction-radiation by an offshore structure (without forward speed) and steady flows, and to the Green function of wave diffraction-radiation at small forward speed, are presented in [4,5]. Another application, to time-harmonic ship waves characterized by the dispersion function (5), is examined here. Only the wave component *φ*^{W} is considered; a complementary detailed study of the near-field component *φ*^{N} will be reported elsewhere.

The wave component *φ*^{W} is defined by (35) as

where *φ*^{iV}*, φ*^{oV}*, φ*^{R} and *φ*^{RF} represent distinct wave components, associated with the dispersion curves located within the regions of the Fourier plane defined by (32), which correspond to inner V waves, outer V waves, ring waves and ring-fan waves. Fourier representations of these wave components are given in the study. The spectrum function *S,* typically associated with a continous or discrete distribution of sources and/or dipoles over the hull of a ship, is arbitrary in these Fourier representations, which therefore provide simple explicit analytical representations of the wave components radiated by an arbitrary (volume, surface, and/or line) distribution of singularities (e.g. sources and/or dipoles), including the special case of a point source, i.e. the free-surface Green function. The integrands of the Fourier integrals defining the wave components *φ*^{iV,}*φ*^{oV}*, φ*^{R} and *φ*^{RF} are continuous, so that these expressions are well suited for numerical evaluation. For purposes of illustration, the wave potential *φ*^{W} in (21) is considered for a simple example spectrum function.

The first and second authors were supported by DTMB's Independent Research program and a DRET research grant, respectively.

[1] Kochin, N.E. ( 1937) *On the wavemaking resistance and lift of bodies submerged in water**,* translated in SNAME Tech. and Res. Bull. 1–8 (1951).

[2] Kochin, N.E. ( 1940) *The theory of waves generated by oscillations of a body under the free surface of a heavy* *incompressible fluid**,* translated in SNAME Tech. and Res. Bull. 1–10 (1952).

[3] Noblesse, F. and Yang, C. ( 1995) *Fourier-Kochin formulation of wave-diffraction-radiation by ships or offshore* *structures**,* Ship Technology Research, 42/3, 115–139.

[4] Noblesse, F. and Chen, X.B. ( 1995) *Decomposition of free-surface effects into wave and near-field components* *,* Ship Technology Research, 42/4, 167–185.

[5] Noblesse, F. and Yang, C. ( 1996) *Fourier representation of near-field free-surface flows**,* Ship Technology Research, 43/1, 19–37.

[6] Campana, E.; Di Mascio, A.; Esposito, P.G.; Lalli, F. ( 1993) *Domain decomposition in free surface viscous flows**,* 6th Intl Conf. Numerical Ship Hydro.

[7] Chen, H.C.; Lin, W.M.; Weems, K.M. ( 1993) *Interactive zonal approach for ship flow including viscous nonlinear and wave effects**,* 6th Intl Conf. Numerical Ship Hydro.