Effective dipolar boundary conditions for dynamic magnetization in thin magnetic stripes

We show that dynamic magnetization at the lateral edges of a thin, axially magnetized magnetic element with finite in-plane size can be described by effective pinning boundary conditions. This effective pinning is of a purely dipolar nature, is not related to the magnetocrystalline surface anisotropy of the magnetic material, and is determined by the inhomogeneity of the dynamic demagnetizing field near the edges of the element. Eigenfunctions and eigenvalues obtained using these effective boundary conditions give quantitative description of the quantized spin wave spectra experimentally observed in long and thin permalloy stripes of a micron-size width.