The dispersion curves defined by the dispersion relation for diffraction-radiation by a ship advancing through regular (time-harmonic) waves, within the classical 3D frequency-domain potential-flow framework, in uniform finite water depth are considered.
The curves, called dispersion curves, defined by the dispersion relation (1) in the Fourier plane (α, β) are of fundamental importance. Indeed, farfield wave patterns are directly defined in terms of the dispersion curves, which are also important for the construction of Green functions; e.g. Noblesse and Yang (2004). In the deep-water limit, the dispersion curves defined by the dispersion relation (1), with tan h(d k) = 1, have been considered in detail in the literature; e.g. Noblesse and Yang (2004). In the more general case of finite water depth considered here, the dispersion curves defined by (1) are significantly more complicated and have not been extensively studied (Chen and Nguyen, 2000).
The dispersion curves related to the dispersion relation (1) can be defined by parametric equations based on the wavenumbers k = ±α for which the dispersion curves intersect the axis β = 0. These parametric equations are given in Noblesse and Yang (2004) for deep water, and can easily be extended to the more general case of finite water depth considered here. The intersection wavenumbers k = ±α provide reference wavenumbers for the parametric representations of the dispersion curves given in Noblesse and Yang (2004). Furthermore, these intersection wavenumbers yield fundamental wavenumbers (related to the longest wave) for the distinct wave systems associated with the separate branches of the dispersion curves related to (1); see Noblesse and Yang (2004). More generally, Tc is a function of the water depth. The regime change that occurs at T = Tc is now considered.
