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We propose a mathematical model for describing radially propagating spinwaves emitted from the core region in a magnetic patch with n vertices in amagnetic vortex state. The azimuthal anisotropic propagation of surface spinwaves (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 tounderstand the spin wave propagation in these systems, we first use an approachbased on geometrical curves called 'hippopedes', however it provides no insightinto the underlying physics. Analytical models rely on generalized expressionsfrom the dispersion relation of SSW with an arbitrary angle betweenmagnetization M and wavenumber k. The derived algebraic expression for theazimuthal dispersion is found to be equivalent to that of the 'hippopede'curves. The fitting curves from the model yield a spin wave wavelength for anygiven azimuthal direction, number of patch vertices and excitation frequency,showing a connection with fundamental physics of exchange dominated surfacespin waves. Analytical results show good agreement with micromagneticsimulations and can be easily extrapolated to any n-corner patch geometry.

Hippopede curves for modelling radial spin waves in an azimuthal graded magnonic landscape