Classical and quantum theory of spin waves in the vortex state of a mesoscopic sub-micron magnetic disk has been developed with account of the finite mass density of the vortex. Oscillations of the vortex core resemble oscillations of a charged string in a potential well in the presence of the magnetic field. Conventional gyroscopic frequency appears as a gap in the spectrum of spin waves of the vortex. The mass of the vortex has been computed that agrees with experimental findings. Finite vortex mass generates a high-frequency branch of spin waves. Effects of the external magnetic field and dissipation have been addressed.