  We propose a mathematical model for describing radially propagating spin
waves emitted from the core region in a magnetic patch with n vertices in a
magnetic vortex state. The azimuthal anisotropic propagation of surface spin
waves (SSW) into the domain, and confined spin waves (or Winter's Magnons, WM)
in domain walls increases the complexity of the magnonic landscape. In order to
understand the spin wave propagation in these systems, we first use an approach
based on geometrical curves called 'hippopedes', however it provides no insight
into the underlying physics. Analytical models rely on generalized expressions
from the dispersion relation of SSW with an arbitrary angle between
magnetization M and wavenumber k. The derived algebraic expression for the
azimuthal dispersion is found to be equivalent to that of the 'hippopede'
curves. The fitting curves from the model yield a spin wave wavelength for any
given azimuthal direction, number of patch vertices and excitation frequency,
showing a connection with fundamental physics of exchange dominated surface
spin waves. Analytical results show good agreement with micromagnetic
simulations and can be easily extrapolated to any n-corner patch geometry.

Mesoscale and Nanoscale Physics