  Galactic nuclei (GNs) are dense stellar environments abundant in
gravitational-wave (GW) sources for LIGO, VIRGO, and KAGRA. The GWs may be
generated by stellar-mass black hole (BH) or neutron star mergers following
gravitational bremsstrahlung, dynamical scattering encounters, Kozai-Lidov type
oscillations driven by the central supermassive black hole (SMBH), or
gas-assisted mergers if present. In this paper, we examine a smoking gun
signature to identify sources in GNs: the GWs scattered by the central SMBH.
This produces a secondary signal, an astrophysical GW echo, which has a very
similar time-frequency evolution as the primary signal but arrives after a time
delay. We determine the amplitude and time-delay distribution of the GW echo as
a function of source distance from the SMBH. Between $\sim10\%-90\%$ of the
detectable echoes arrive within $\sim(1-100)M_6\,\mathrm{sec}$ after the
primary GW for sources between $10-10^4$ Schwarzschild radius, where
$M_6=M_{\rm SMBH,z}/(10^6\,\mathrm{M}_{\odot})$, and $M_{\rm SMBH,z}$ is the
observer-frame SMBH mass. The echo arrival times are systematically longer for
high signal-to-noise ratio (SNR) primary GWs, where the GW echo rays are
scattered at large deflection angles. In particular, $\sim10\%-90\%$ of the
distribution is shifted to $\sim(5-1800)M_6\,\mathrm{sec}$ for sources, where
the lower limit of echo detection is $0.02$ of the primary signal amplitude. We
find that $\sim5\%-30\%$ ($\sim1\%-7\%$) of GW sources have an echo amplitude
larger than $0.2-0.05$ times the amplitude of the primary signal if the source
distance from the SMBH is $50$ ($200$) Schwarzschild radius. Non-detections can
rule out that a GW source is near an SMBH.
