  Although spin cycloids and helices are quite common, remarkably little is
known about the normal modes of a spin cycloid or helix with finite length on a
discrete lattice. Based on simple one-dimensional lattice models, we
numerically evaluate the normal modes of a spin cycloid or helix produced by
either Dzyaloshinskii-Moriya (DM) or competing exchange (CE) interactions. The
normal modes depend on the type of interaction and on whether the
nearest-neighbor exchange is antiferromagnetic (AF) or ferromagnetic (FM). In
the AF/DM and FM/DM cases, there is only a single Goldstone mode; in the AF/CE
and FM/CE cases, there are three. For FM exchange, the spin oscillations
produced by non-Goldstone modes contain a mixture of tangential and transverse
components. For the DM cases, we compare our numerical results with analytic
results in the continuum limit. Examples are given of materials that fall into
all four cases.
