How black hole ringdown signals actually build up during a plunge

When a particle spirals into a black hole, the gravitational wave signal's oscillatory 'ringdown' phase arises from the excitation of quasinormal modes — the black hole's natural vibration frequencies. By carefully analyzing the high-frequency behavior of these mode solutions, the authors identify a 'bounce radius' that divides the signal into two regimes: outside it, successive overtones interfere destructively; inside, they align in phase and the mode sum converges quickly. The resulting waveform matches the oscillatory component accurately from shortly after peak emission, providing a rigorous theoretical basis for ringdown modeling in gravitational wave astronomy.