The purpose of the paper is to obtain an analytic solution of the boundary-value problem for the system of nonlinear partial differential equations that model magnetohydrodynamic perturbations in a layer of perfect electrically conducting rotating fluid bounded by space- and time-varying surfaces with due account of the dissipative factors of the magnetic field diffusion, the inertia forces, and the Coriolis force. We construct an exact solution of the reduced nonlinear equations that describes the propagation of waves of finite amplitude in an infinite horizontally extended electrically conducting fluid when the surface bounding the layer has approximatively constant gradient over distances of the order of the wavelength.