  In this work we derive the interface exchange boundary conditions for the
classical linear dynamics of magnetization in ferromagnetic layers with the
interface Dzyaloshinskii-Moriya interaction (IDMI). We show that IDMI leads to
pinning of dynamic magnetization at the interface. An unusual peculiarity of
the IDMI-based pinning is that its scales as the spin-wave wave number. We
incorporate these boundary conditions into an existing numerical model for the
dynamics of the Damon-Eshbach spin wave in ferromagnetic films. IDMI affects
the dispersion and the frequency non-reciprocity of the traveling Damon-Eshbach
spin wave. For a broad range of film thicknesses L and wave numbers the results
of the numerical simulations of the spin wave dispersion are in a good
agreement with a simple analytical expression which shows that the contribution
of IDMI to the dispersion scales as 1/L, similarly to the effect of other types
of interfacial anisotropy. Suggestions to experimentalists how to detect the
presence of IDMI in a spin wave experiment are given.
