© 2015 AIP Publishing LLC. The dynamic behavior of perpendicularly magnetized permalloy circular rings is systematically investigated as a function of film thickness using broadband field modulated ferromagnetic resonance spectroscopy. We observed the splitting of one spin wave mode into a family of dense resonance peaks for the rings, which is markedly different from the single mode observed for continuous films of the same thickness. As the excitation frequency is increased, the mode family observed for the rings gradually converges into one mode. With the increase in the film thickness, a sparser spectrum of modes is observed. Our experimental results are in qualitative agreement with the dynamic micromagnetic simulations.