Propagation of Rayleigh surface waves across a large-amplitude grating

Abstract

The dispersion relation for a Rayleigh surface wave propagating normal to the grooves of a diffraction grating has been obtained by two different methods. The first is based on the use of the Rayleigh hypothesis. The second uses the extinction theorem form of Green's theorem to obtain an exact pair of homogeneous linear integral equations for the boundary values of the elastic displacement field, from the solvability condition for which the Rayleigh-wave dispersion relation is obtained. When the latter pair of equations are solved by Fourier-series expansions, the dispersion relation obtained is identical with the one obtained on the basis of the Rayleigh hypothesis, in the case that the grating profile function is an even function of its argument. Numerical solutions of the dispersion relation are obtained for a sinusoidal profile and for a symmetric sawtooth profile. For the former profile, solutions are found to be convergent for corrugation strengths far beyond the value at which the Rayleigh hypothesis is known to become invalid for the scattering of a scalar plane wave from a hard sinusoidal surface. For the sawtooth profile (for which in the aforementioned problem the Rayleigh hypothesis is invalid due to the nonanalyticity of the profile) convergent results are here obtained for appreciable values of the corrugation strength. These results indicate that the Rayleigh hypothesis can be used for the calculation of the dispersion curve for a Rayleigh wave propagating across a large-amplitude grating, even when the grating profile function is not an analytic function of its argument. The dispersion curves obtained possess gaps for values of the one-dimensional wave vector, characterizing the propagation of the Rayleigh wave, corresponding to the boundaries of the one-dimensional Brillouin zones defined by the period of the grating. An analytic expression is obtained for the form of the dispersion curve in the vicinity of each gap, in the small corrugation limit, for an arbitrary grating profile. Comparison of the predictions of this analytic result with the exact numerical results allows the limits of validity of the small-amplitude approximation to be determined.