A new depth-integrated equation is derived to model a time-harmonic motion of small-amplitude waves in water of variable depth. The new equation, which is referred to as the complementary mild-slope equation here, is derived from Hamilton's principle in terms of stream function. In the formulation, the continuity equation is satisfied exactly in the fluid domain. Also satisfied exactly are the kinematic boundary conditions at the still water level and the uneven sea bottom. The numerical results of the present model are compared to the exact linear theory and the existing mild-slope equations that have been derived from the velocity-potential formulation. The computed results give better agreement with those of the exact linear theory than the other mild-slope equations. Comparison shows that the new equation provides accurate results for a bottom slope up to 1.

This paper presents a numerical method for solving linearized water-wave problems with oscillatory time dependence. Specifically it considers the diffraction problem for oblique plane waves incident upon an infinitely long fixed cylinder on the free surface. The numerical method is based on a variational principle equivalent to the linearized boundary-value problem. Finite-element techniques are used to represent the velocity potential; and the variational principle is used to determine the unknown coefficients in the solution throughout the fluid domain. To illustrate this method, reflexion and transmission coefficients and the diffraction forces and moment are computed for oblique waves incident upon a vertical flat plate, a horizontal flat plate and rectangular cylinders, where the comparison is made with the existing results by others. Also considered is the associated sinuous forced-motion problem, where comparison is made with the results for a circular cylinder obtained by Bolton & Ursell (1973).

This paper presents numerical results for the added-mass and damping coefficients of semi-submerged two-dimensional heaving cylinders in water of finite depth. A simple proof is given which shows that the added mass is bounded for all frequencies in water of finite depth. The limits of the added-mass and damping coefficients are studied as the frequency tends to zero and to infinity. A new formulation valid in the low-frequency limit is constructed by using a perturbation expansion in the wavenumber parameter. For the limiting cases, dual extremum principles are used, which consist of two variational principles: a minimum principle for a functional and a maximum principle for a different but related functional. These two functionals are used to obtain lower and upper bounds on the added mass in the limiting cases. However, the functionals constructed (Bai & Yeung 1974) for the general frequency range (excluding the limiting cases) have neither a minimum nor a maximum. In this case, the approximate solution cannot be proved to be bounded either below or above by the true solution. To illustrate these methods, the added-mass and damping coefficients are computed for a circular cylinder oscillating in water of several different depths. Results are also presented for rectangular cylinders with three different beamdraft ratios at several water depths.

Recently, the authors have derived a new approximate model for the nonlinear water waves, the Irrotational Green-Naghdi (IGN) model. In this paper, we first derive the IGN equations applicable to variable water depth, then perform numerical tests to show whether and how fast the solution of the IGN model converges to the true solution as its level increases. The first example given is the steady solution of the progressive waves of permanent form, which includes the small amplitude sinusoidal wave, the solitary wave and the nonlinear Stokes wave. The second example is the run-up of a solitary wave on a vertical wall. The last example is the shoaling of a wave train over a sloping beach. In each numerical test, the self-convergence of the IGN model is shown first. Then the converged solution is compared to the known analytic solutions and/or solutions of other approximate models such as the KdV and the Boussinesq equations.

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