can propagate with unchanged shape. They should be valid for transcritical and supercritical speeds. In the subcritical speed range, not all wave components are long enough compared to the water depth. Therefore, the classical Boussinesq equations have to be modified. One established method for this is to improve the dispersion relation of the linearized Boussinesq equations by comparison with the linear wave theory. It can be theoretically shown that the modified Boussinesq equations used in the present study are valid for a ratio of wave length to water depth down to a value of 2, practically in the deep-water range. However, due to the influence of the high-oder terms the rapid upwind and high-resolution schemes for the Airy shallow-water wave equations, implemented by Jiang (1996), can not be applied for the Boussinesq equations. A new slow, but numerically more stable, implicit scheme is implemented to solve the initial/boundary value problem governed by the modified Boussinesq equations. It contains:

  • Crank-Nicholson scheme for the time and space discretizations

  • Approximation of the state values for the nonlinear terms by means of Taylor expansion

  • Iterative solution of the linear algebraic equation system

  • Overrelaxation to accelerate the convergence process

  • Local and global filtering to suppress the numerical oscillations

The newly developed computer programs are used to simulate the wave generation by a Series 60 hull moving uniformly in a shallow water channel. A slender-body theory along with the technique of asymptotic expansions is applied to approximate the flow generated by the hull. Both calculated integral quantities, such as wave resistance, sinkage and trim, as well as field quantities, such as wave profiles in the tank, are compared with those from recent measurements in the Duisburg Shallow Water Tank.


General Description

For describing the flow generated by a ship at a uniform speed of V in a shallow water of constant water depth ho, a Cartesian coordinate system Oxyz is used, see Fig. 1. The plane Oxy is on the quiet free surface with the axes x in the direction of ship motion. The center O is on the intersection line of the midship section and the longitudinal centerplane, rendering the involved problem symmetric to the plane Ozx. The axis z is positive upward. The ship is free to heave and pitch; the midpoint sinkage is designated by s (positive downward) and the pitch angle by θ (positive bow-down).

Fig. 1 Schematic of the problem and coordinate system

By virtue of this symmetry only the half fluid-domain on the starboard side of the ship is described in the following. On the assumption that the fluid is incompressible and inviscid, the irrotational flow generated by the moving ship can be described by a potential Φ governed by the Laplace equation,


in the whole fluid domain, and by the following boundary conditions: first, kinematic and dynamic conditions,



respectively, on the free surface z=ζ(x, y, t), where g is the acceleration due to gravity; second, no-flux condition,


on the hull-surface F(x, y, z, t)=0; third, no-flux condition,


on the water bottom z=−ho; fourth, no-flux condition,


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