Propagation of a surface wave near a randomly rough surface

A generalisation of the problem on the propagation of a surface elastic Rayleigh wave near a free surface obtained by continuous deformation of the initial plane is considered. The set of possible realisations of the surface is, on average, equivalent to a plane, and the dispersion is a constant. The smallness of a dimensionless parameter, the gradient to a surface, is assumed, which causes the presence of small fluctuations in all field quantities. The effective boundary conditions on a plane boundary are obtained. From the condition for the existence of non-zero solutions, the generalised Rayleigh equation is found for the case of an uneven boundary containing a parameter of a dimensionless dispersion of the gradient to a surface. Roots of the dispersion equation are numerically found depending on the Poisson’s ratio and dispersion. The influence of the dispersion of surface roughness is manifested in the appearance of an additional root under the condition that the ratio of the Rayleigh wave velocity to the transverse velocity is less than unity. The second root corresponds to the appearance of a wave slower than the Rayleigh one, the amplitude of which also decreases with depth. Physically acceptable solutions can only exist for a dispersion value of less than 0.09 in the range of varying of material properties from solid to rubbery.