2D-parametric model is used to simulate waves under Tropical Cyclones (TCs). Set of equations describing either wind waves development and swell evolution, is solved using method of characteristics. Wave-rays patterns provide efficient visualization on how wave trains develop and travel through TC varying wind field and leave storm area as swell. The superposition of wave-trains rays exhibits coherent spatial patterns of significant wave height, peak wavelength and direction, depending on TC characteristics, - maximal wind speed (um), radius (Rm), and translation velocity (V). Group velocity resonance leads to appearance of waves with abnormal energy between the TC right and front sectors, further outrunning as swell through the TC front sector. Yet, when TC translation velocity exceeds a threshold value, waves cannot reach group velocity resonance, and travelling backwards, form a wake of swell systems trailing the forward moving TC. 2D-parametric model solutions are parameterized using 2D self-similar universal functions. Comparisons between self-similar solutions and measurements, demonstrate excellent agreement to warrant their use for scientific and practical applications. Self-similar solutions provide immediate estimates of azimuthal-radial distributions of wave parameters under TCs, solely characterized by arbitrary sets of um, Rm and V conditions. Self-similar solutions clearly divide TCs between slow TCs fulfilling conditions Rm/Lcr>1, and fast TCs corresponding to Rm/Lcr <1, where Lcr is a critical fetch. The region around Rm/Lc = 1 corresponds to the group velocity resonance conditions, leading to the largest possible waves generated by a TC.
