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

The wavefront of the radio signal emitted by cosmic ray air showers