Nonlinear surface waves on an elastic solid

The effect of quadratic elastic nonlinearity on the propagation of surface Rayleigh waves on an isotropic elastic solid is examined. Using the method of multiple scales an approximate solution is obtained which is uniformly valid in both spatial directions as well as in time. An arbitrary wave profile is considered and an integro-differential equation is derived for the Fourier transform of the displacement on the boundary. In the case of a quasi-monochromatic wave explicit expressions are derived for the variations of the amplitudes of the fundamental and second and third harmonics along the boundary.