Analyzing measurements of the LOPES antenna array together with corresponding CoREAS simulations for more than 300 measured events with energy above 1017 eV and zenith angles smaller than 45̂, 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 ≳ 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 experime ... mehrntally achieved precision for the shower maximum is limited by measurement uncertainties to approximately 140 g/c 2. But the simulations indicate that under better conditions this method might yield an accuracy for the atmospheric depth of the shower maximum, Xmax, better than 30 g/c 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.