  We develop a field theory to study the dynamics of long wavelength exchange
spin wave excitations on honeycomb nanoribbons characterized by armchair edge
boundaries and the N\'eel antiferromagnetic ordering state. Appropriate
boundary conditions are established by requiring that the bulk and edge spins
precess with the same frequency for any given spin wave eigenmode in these
systems. A set of characteristic boundary equations, common for bulk and edge
spin wave modes, are hence derived. The equations of motion for the spin
dynamics are then solved to determine the propagating and evanescent exchange
spin wave modes. We prove in general that the bulk spin wave dynamics is
discretized due to the finite width of the nanoribbon. For an isotropic
magnetic nanoribbon, the Dirac cone is reduced to a single linear dispersion
curve due to this discretization. The number and wavelengths of allowed bulk
modes for isotropic and anisotropic nanoribbons are determined from the derived
characteristic boundary equations. As witnessed by our numerical results for
different examples it is shown that the characteristics of these modes depend
on the width of the nanoribbon and its antiferromagnetic anisotropy. Further,
anisotropic nanoribbons, even those with the slightest anisotropy, present
evanescent modes with non-linear dispersion relations. The spatial variation of
the amplitudes of the evanescent exchange spin waves across the finite widths
of the nanoribbons, is found to be strongly dependent on the system magnetic
anisotropy and its width. The developed theoretical approach is general and can
be applied for nanoribbons with all types of boundary edges.
