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The propagation of waves of water on the surface is characterized by various mathematical models. In this work, we adopt the simplest equation method as well as the Kudryashov's new function method, to a variety of some nonlinear models of surface wave propagation to extract their abundant solitary wave structures. These nonlinear models include the Benjamin‐Bona‐Mahony (BBM) equation, the Ostrovsky (OS) equation, the Ostrovsky‐Benjamin‐Bona‐Mahony (OS‐BBM) equation, and the Boussinesq system of equations. Based on the solutions of two distinct auxiliary equations, various wave solutions are successfully obtained. In addition, the geometric interpretations of the obtained solutions are analyzed. To illuminate the physical nature of the obtained solutions of this paper, some graphical representations (numerical simulations) of the acquired solutions have been depicted in two‐dimensional density, two‐ and three‐dimensional graphs. Furthermore, a comparison between the obtained results of this paper and the solutions for the considered models obtained in the literature is also provided. It is shown that the used approaches are very effective and powerful strategies for a wider class of nonlinear partial differential equations in physical problems. [ABSTRACT FROM AUTHOR]