An investigation is carried out into the effect on wave propagation of an ice sheet of varying thickness floating on water of varying depth, in three dimensions. By deriving a variational principle equivalent to the governing equations of linear theory and invoking the mild-slope approximation in respect of the ice thickness and water depth variations, a simplified form of the problem is obtained from which the vertical coordinate is absent. Two situations are considered: the scattering of flexural–gravity waves by variations in the thickness of an infinite ice sheet and by depth variations; and the scattering of free-surface gravity waves by an ice sheet of finite extent and varying thickness, again incorporating arbitrary topography. Numerical methods are devised for the two-dimensional versions of these problems and a selection of results is presented. The variational approach that is developed can be used to implement more sophisticated approximations and is capable of producing the solution of full linear problems by taking a large enough basis in the Rayleigh–Ritz method. It is also applicable to other situations that involve wave scattering by a floating elastic sheet.