Extended mild-slope equations for the propagation of small-amplitude water waves over variable bathymetry regions, recently proposed by Massel (1993) and Porter & Staziker (1995), are shown to exhibit an inconsistency concerning the sloping-bottom boundary condition, which renders them non-conservative with respect to wave energy. In the present work, a consistent coupled-mode theory is derived from a variational formulation of the complete linear problem, by representing the vertical distribution of the wave potential as a uniformly convergent series of local vertical modes at each horizontal position. This series consists of the vertical eigenfunctions associated with the propagating and all evanescent modes and, when the slope of the bottom is different from zero, an additional mode, carrying information about the bottom slope. The coupled-mode system obtained in this way contains an additional equation, as well as additional interaction terms in all other equations, and reduces to the previous extended mild-slope equations when the additional mode is neglected. Extensive numerical results demonstrate that the present model leads to the exact satisfaction of the bottom boundary condition and, thus, it is energy conservative. Moreover, it is numerically shown that the rate of decay of the modal-amplitude functions is improved from O(n⁻²), where n is the mode number, to O(n⁻⁴), when the additional sloping-bottom mode is included in the representation. This fact substantially accelerates the convergence of the modal series and ensures the uniform convergence of the velocity field up to and including the boundaries.

In this paper we couple collocated Boundary Element Methods (BEM) with unstructured analysis-suitable T-spline surfaces for solving a linear Boundary Integral Equation (BIE) arising in the context of a ship-hydrodynamic problem, namely the so-called Neumann–Kelvin problem, following the formulation by Brard (1972) and Baar and Price (1988). The local-refinement capabilities of the adopted T-spline bases, which are used for representing both the geometry of the hull and approximating the solution of the associated BIE, in accordance with the Isogeometric concept proposed by Hughes et al. (2005), lead to a solver that achieves the same error level for many fewer degrees of freedom as compared with the corresponding NURBS-based Isogeometric-BEM solver recently developed in Belibassakis et al. (2013). In this connection, this paper makes a step towards integrating modern CAD representations for ship-hulls with hydrodynamic solvers of improved accuracy and efficiency, which is a prerequisite for building efficient ship-hull optimizers