In this paper, a variety of problems concerned with the interaction of water waves with fixed horizontal plates lying on the surface of a fluid are investigated. Firstly, solutions are presented to the problem of the scattering of incident waves by: (i) infinitely-long plates of constant finite width (often referred to as the two-dimensional `finite dock problem') and (ii) finite plates which are either rectangular or parallelogram-shaped. Secondly, hydrodynamic coefficients due to forced motions of plates are also considered. Finally, eigenvalue problems associated with free oscillations of the surface in long channels of uniform width and finite rectangular holes in an otherwise infinite rigid plate covering the surface are considered. A common method of solution is applied to all problems which involves using Fourier transforms to derive integral equations for unknown potentials over finite regions of space occupied by either plates or the free surface. Integral equations are converted, using the Galerkin method, into second-kind infinite systems of algebraic equations. In each problem numerical approximations to the solutions are found to converge rapidly with increasing truncation size of the infinite system making this approach both numerically efficient and accurate. Some comparisons with existing results are made, and new results for finite plates are demonstrated.