  We present a new method for probing the hadronic interaction models at
ultra-high energy and extracting details about mass composition. This is done
using the time profiles of the signals recorded with the water-Cherenkov
detectors of the Pierre Auger Observatory. The profiles arise from a mix of the
muon and electromagnetic components of air-showers. Using the risetimes of the
recorded signals we define a new parameter, which we use to compare our
observations with predictions from simulations. We find, firstly,
inconsistencies between our data and predictions over a greater energy range
and with substantially more events than in previous studies. Secondly, by
calibrating the new parameter with fluorescence measurements from observations
made at the Auger Observatory, we can infer the depth of shower maximum for a
sample of over 81,000 events extending from 0.3 EeV to over 100 EeV. Above 30
EeV, the sample is nearly fourteen times larger than currently available from
fluorescence measurements and extending the covered energy range by half a
decade. The energy dependence of the average depth of shower maximum is
compared to simulations and interpreted in terms of the mean of the logarithmic
mass. We find good agreement with previous work and extend the measurement of
the mean depth of shower maximum to greater energies than before, reducing
significantly the statistical uncertainty associated with the inferences about
mass composition.
