Wave propagation along periodic layers

Solutions for waves travelling transversely in a layered medium satisfy an exact recurrence relation when the transit time across each of the layers is the same. Using such solutions in an alternating-direction splitting method, high accuracy solutions can also be computed for wave propagation parallel to the layers. An asymptotic analysis uncovers the remarkable result that dispersion of waves parallel to the layers is very similar to the dispersion observed in the transverse case. This result is confirmed by the numerical solutions. The results are in general agreement with similar observations reported in the literature using di↵erent numerical and analytical methods.