Low-finesse scattering and non-stationary dispersive dynamics of gravitational wave echoes

Abstract We study environmental echoes induced by a weak potential barrier outside a Schwarzschild black hole. In the low-finesse limit, the time domain response is governed by a sequence of transient wave packets formed by finite round-trip scattering, rather than steady state cavity modes. We establish quantitative criteria for the breakdown of the steady state resonance picture, dictated by frequency domain spectral aliasing and time domain truncation from the black hole power law tail. Based on non-stationary dispersive dynamics, we analytically derive the arrival time gliding, central frequency drift, and dispersion driven asymmetric tails of these echoes. Accordingly, we construct a five-parameter analytical reconstruction whose overlap with the finite difference time domain extracted first echo is nearly identical to that obtained from the full transfer-function reconstruction. Our results demonstrate that early low-finesse environmental echoes must be theoretically modeled as non-stationary transient scattering signals.