Surface waves in water or granular layers and on the surface of weakly cohesive upper-lying soils are studied. A one-dimensional perturbed wave equation is derived for these waves. It is shown that the waves may be excited due to local topographies and a vertical excitation. The velocity of the waves depends on the geometry of the layer, the mechanical properties of the material and the vertical forced acceleration. Approximate solutions of the equation are presented which take into account resonant, nonlinear, dispersive, dissipative, topographic and parametric effects. The solutions describe unfamiliar waves which cannot be classified as soliton-, cnoidal-, shock- or breather-type waves. In particular, the solutions describe spatiotemporally oscillating, localized, nonlinear, surface waves which possess properties of both standing waves and travelling waves. They are not d'Alembert-type waves. Different wave patterns are yielded by the solutions in the x-t plane. Topographic and parametric effects are analysed. Sometimes these effects are dependent. The topographic effect explains some unexpected results of both experiments and earthquakes. An observation of Charles Darwin is discussed. Perhaps the solutions describe waves which may be in different wave fields of Nature.