Spiral galaxies have been observed for over 150 years, but so far no satisfactory explanation for the origin of their beautiful spiral patterns has been provided. Most theorists expect them to be a consequence of gravitationally-driven, collective oscillations of the stellar system, which is supported by the evidence that recurrent spiral patterns appear spontaneously in large numerical simulations. While the physics of small-amplitude spiral wave packets is well-established, most previous attempts to calculate modes have foundered because spiral waves are strongly damped at resonances in smooth stellar disks. I demonstrate that the non-linear scattering of stars by one spiral wave creates impedance changes in the stellar disk from which new waves can reflect, allowing true standing waves to be established and leading to a recurrent cycle of spiral instabilities.