  Analyzing measurements of the LOPES antenna array together with corresponding
CoREAS simulations for more than 300 measured events with energy above
$10^{17}\,$eV and zenith angles smaller than $45^\circ$, we find that the radio
wavefront of cosmic-ray air showers is of approximately hyperbolic shape. The
simulations predict a slightly steeper wavefront towards East than towards
West, but this asymmetry is negligible against the measurement uncertainties of
LOPES. At axis distances $\gtrsim 50\,$m, the wavefront can be approximated by
a simple cone. According to the simulations, the cone angle is clearly
correlated with the shower maximum. Thus, we confirm earlier predictions that
arrival time measurements can be used to study the longitudinal shower
development, but now using a realistic wavefront. Moreover, we show that the
hyperbolic wavefront is compatible with our measurement, and we present several
experimental indications that the cone angle is indeed sensitive to the shower
development. Consequently, the wavefront can be used to statistically study the
primary composition of ultra-high energy cosmic rays. At LOPES, the
experimentally achieved precision for the shower maximum is limited by
measurement uncertainties to approximately $140\,$g/cm$^2$. But the simulations
indicate that under better conditions this method might yield an accuracy for
the atmospheric depth of the shower maximum, $X_\mathrm{max}$, better than
$30\,$g/cm$^2$. This would be competitive with the established air-fluorescence
and air-Cherenkov techniques, where the radio technique offers the advantage of
a significantly higher duty-cycle. Finally, the hyperbolic wavefront can be
used to reconstruct the shower geometry more accurately, which potentially
allows a better reconstruction of all other shower parameters, too.
