  We propose a mathematical model for describing propagating confined modes in
domain walls of intermediate angle between domains. The proposed model is
derived from the linearised Bloch equations of motion and after reasonable
assumptions, in the scenario of a thick enough magnetic patch, are accounted.
The model shows that there is a clear dependence of the local wavenumber of the
confined spin wave on the local angle of the wall and excitation frequency
used, which leads to the definition of a local index of refraction in the wall
as a function of such angle and frequency. Therefore, the model applies to 1-D
propagating modes, although it also has physical implications for 2-D scenarios
where a domain wall merges with a saturated magnetic region. Micromagnetic
simulations are in good agreement with the predictions of the model and also
give insight on the effects of curved finite structures may have on the
propagating characteristics of spin waves in domain walls.
