The replacement of a layered medium by a homogeneous transversely isotropic medium is justified if all wavelengths are sufficiently long. High-resolution techniques, the increasing use of shear waves, and attention to stratigraphic detail require a quantitative evaluation of what is sufficiently long, as well as a study of what happens for wavelengths between that limit and the limit of resolution. Such information can be obtained through a numerical evaluation of the general dispersion equation: one obtains the frequency as a function of the spatial wave vector k. The phase velocity is the v = omega k/(k.k) and the group velocity g = grad (sub kappa ) omega . In general, both dispersion and anisotropy are to be expected. This method has been applied to SH-waves in periodically layered media in general, and the dispersion equation has been evaluated numerically for two representative media. The most significant result is that the long-wavelengths approach, i.e., a nondispersive transversely isotropic replacement medium, is strictly valid for wavelengths larger than three times the spatial period of layering. For small angles against the vertical, dispersion for shorter wavelength is significant. However, for directions making an angle of more than about 30 degrees with the vertical, dispersion sets in at much shorter wavelengths and is in general much more gentle.--Modified journal abstract.