Rayleigh Waves in a Homogeneous Isotropic Half-Plane with a Periodic Edge

The existence of Rayleigh waves propagating along the periodic boundary of a homogeneous isotropic half-plane with an arbitrary edge profile is proved. It is shown that a periodic family of cracks perpendicular to the straight boundary of the half-plane, with an increase in the lengths, is capable of forming an arbitrary predetermined number of linearly independent localized waves in the low-frequency range of the spectrum including the standing ones, which transfer no energy along the edge. The results are obtained by means of variational and asymptotic methods of spectral analysis of a model problem in a half-strip with a curved end and with quasi-periodicity conditions on the lateral sides.