The paper is concerned with the propagation of shear horizontal surface waves (SHSW) in semi-infinite elastic media with vertically periodic continuous and/or discrete variation of material properties. The existence and spectral properties of the SHSW are shown to be intimately related to the shape of the properties variation profile. Generally, the SHSW dispersion branches represent randomly broken spectral intervals on the (
ω
,
k
) plane. They may, however, display a particular regularity in being confined to certain distinct ranges of slowness
s
=
ω
/
k
, which can be predicted and estimated directly from the profile shape. The SHSW spectral regularity is especially prominent when the material properties at the opposite edge points of a period are different. In particular, a unit cell can be arranged so that the SHSW exists within a single slowness window, narrow in the measure of material contrast between the edges, and does not exist elsewhere or vice versa. Explicit analysis in the (
ω
,
k
) domain is complemented and verified through the numerical simulation of the SH wave field in the time–space domain. The results also apply to a longitudinally periodic semi-infinite strip with a homogeneous boundary condition at the faces.
Acousto-Optic Diffraction by Shear Horizontal Surface Acoustic Waves in 36° Rotated Y-Cut X-Propagation Lithium Tantalate