  Strong gravitational lensing of gravitational wave sources offers a novel
probe of both the lens galaxy and the binary source population. In particular,
the strong lensing event rate and the time delay distribution of
multiply-imaged gravitational-wave binary coalescence events can be used to
constrain the mass distribution of the lenses as well as the intrinsic
properties of the source population. We calculate the strong lensing event rate
for a range of second (2G) and third generation (3G) detectors, including
Advanced LIGO/Virgo, A+, Einstein Telescope (ET), and Cosmic Explorer (CE). For
3G detectors, we find that {$\sim0.1\%$} of observed events are expected to be
strongly lensed. We predict detections of {$\sim 1$} lensing pair per year with
A+, and {$\sim 50$} pairs {per year} with ET/CE. These rates are highly
sensitive to the characteristic galaxy velocity dispersion, $\sigma_*$,
implying that observations of the rates will be a sensitive probe of lens
properties. We explore using the time delay distribution between
multiply-imaged gravitational-wave sources to constrain properties of the
lenses. We find that 3G detectors would constrain $\sigma_*$ to {$\sim21\%$
after 5 years}. Finally, we show that the presence or absence of strong lensing
{within the detected population} provides useful insights into the source
redshift and mass distribution out to redshifts beyond the peak of the star
formation rate, which can be used to constrain formation channels and their
relation to the star formation rate and delay time distributions for these
systems.
