The nonlinear equations that underlie the analysis of classical Rayleigh waves are derived for the two-dimensional case of nonlinear elastic deformation described by the Murnaghan model. In addition to the case of presence of both geometrical and physical nonlinearities, two special cases are considered where one only type of nonlinearity is taken into account. It is shown that unlike the one-dimensional problems for plane waves where only three types of nonlinear interaction should be allowed for, the two-dimensional problems should include 24 types of nonlinear interaction. In the case of geometrical nonlinearity alone, a preliminary analysis of the nonlinear equations is carried out. Second-order equations are derived. The second approximation includes the second harmonics of the wave itself and its attenuating amplitude and is nonlinearly dependent on the initial amplitude of the Rayleigh wave and linearly increasing with the distance traveled by the wave Keywords: nonlinear elastic two-dimensional plane problem, Murnaghan model, nonlinear Rayleigh wave, second-order wave equations, second harmonicIntroduction. The Rayleigh wave [21] is classed among surface waves and its linear approximation is well studied [8,9].Linear Rayleigh waves are addressed in many fundamental theoretical studies [1, 2, 10, 13-15] which date back to 1885 when the theoretical solution for surface waves was first obtained [21]. Various approaches and methods were used in these studies. Currently, the Stroh formalism [30] is a well-tested method for studying surface waves.Later, Rayleigh waves were studied theoretically for various models (different from the linear elastic model). For example, Rayleigh waves in materials with initial (residual) stresses were analyzed in [3]. The microstructural elastic mixture model was used for this purpose in [7,22].Rayleigh waves were experimentally studied not only by simulation in laboratory [19], but also in natural conditions during earthquakes because Rayleigh waves are used for seismic monitoring [12]. Experimental studies of acoustic surface waves attract interest due to the wide range of applications of their properties.A quite extensive bibliography on nonlinear Rayleigh waves can be found in the old (1974) work [31] by Canadian scientists, the first of the studies [16-18, 32, 33] (1992-2008) by Zabolotskaya with coauthors, and Sgoureva-Philippakos' dissertation [29] (1999).Since nonlinearity in elasticity problems can be described in many ways, it makes sense to note that the constitutive equations in the first of the above-cited publications are a quadratic relationship among stresses, displacements, and displacement gradient in the form of an infinite series of "harmonics" of a linear Rayleigh wave and are solved numerically. The second publication (and the subsequent series of publications) follows the Hamiltonian formalism which incorporates the same quadratic nonlinearity, whereas the constitutive equations used in the third publication include the nonlinear John potential.

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A Rayleigh wave propagating along the boundary surface of an elastic half-space whose nonlinear deformation is described by the Murnaghan model is analyzed. Three simple equations of nonlinear motion for displacement and six new nonlinear wave equations for potentials are proposed. New approximate (first two approximations) solutions are obtained and discussed. It is shown that the second approximation includes the second harmonics of both traveling harmonic wave and its amplitude decreasing with depth and nonlinearly depends on the initial amplitude of the Rayleigh wave. The geometrically linear and nonlinear boundary conditions are written, and their difference is shown. Their role in the analysis of Rayleigh wave is discussed. A new nonlinear equation for the Rayleigh wave number is derived and analyzed Keywords: Murnaghan potential, quadratic nonlinear Rayleigh wave, geometrical and physical nonlinearities, linear and nonlinear boundary conditions, new nonlinear equation for Rayleigh wave numberIntroduction. The results discussed here are related to a two-dimensional model of nonlinear elastic deformation described by the Murnaghan potential. The publication [12] discusses three alternative equations of motion within the framework of this model: (1) deformation is both geometrically and physically nonlinear, (2) deformation is only geometrically nonlinear (strains are not infinitesimal, and the constitutive equations are linear), and (3) deformation is only physically nonlinear (strains are infinitesimal, and the constitutive equations are nonlinear). In some cases, the equations of motion can be simplified by neglecting the nonlinear interaction of displacements (such a simplification is, for example, used in solving the nonlinear equations of motion with the method of slowly varying amplitudes [5,9,11]):