We report on the generation and confinement of short-wavelength spin waves in a continuous film with periodically modulated magnetic anisotropy. The concept, which is demonstrated for strain-coupled Co40Fe40B20/BaTiO3 heterostructures, relies on abrupt rotation of magnetic anisotropy at the boundaries of magnetic stripe domains. In combination with an external bias field, this modulation of magnetic anisotropy produces a lateral variation of the effective magnetic field, leading to local spin-wave excitation when irradiated by a microwave magnetic field. In domains with small effective field, spin waves are perfectly confined by the spin gap in neighboring domains. In contrast, standing spin waves in domains with large effective field radiate into neighboring domains. Using micromagnetic simulation, we show that the wavelength of emitted spin waves is tunable from a few micrometers down to about 100 nm by rotation of the bias field. Importantly, the orientation of the wave front remains fixed. We also demonstrate that dynamic fluctuations of the effective magnetic field produce exchange-dominated spin waves at single anisotropy boundaries. The multiferroic heterostructures presented here enable the use of global excitation fields from a microwave antenna to emit tunable spin waves.