Abstract Two-dimensional water wave problems are investigated in an environment with a flat or sloping rigid seabed adopting a continuous Rankine source method. All the fluid domain surfaces, that is the free, body and seabed surfaces, are discretized using continuous panels. These panels are positioned exactly on the fluid boundary surfaces and no desingularization technique is required. A new seabed source panel distribution method is developed to accommodate both symmetric and asymmetric seabed profiles. To validate the numerical model comparisons are made with published findings from other mathematical models and experimental data. The presence of a sloped seabed alters the symmetry of the fluid domain, causing wave reflection and shoaling, and therefore, significantly affects the hydrodynamic characteristics of water wave problems. The influence of these topographies on the responses in all three degrees of freedom (heave, sway and roll) of a rigid floating body are investigated and discussed accounting for wave radiation and diffraction problems.