The ARIANNA hexagonal radio array (HRA) is an experiment in its pilot phase designed to detect cosmogenic neutrinos of energies above 10^16 eV. The most neutrino-like background stems from the radio emission of air showers. This article reports on dedicated efforts of simulating and detecting the signals of cosmic rays. A description of the fully radio self-triggered data-set, the properties of the detected air shower signals in the frequency range of unit[100-500]{MHz} and the consequences for neutrino detection are given. 38 air shower signals are identified by their distinct waveform characteristics, are in good agreement with simulations and their signals provide evidence that neutrino-induced radio signals will be distinguishable with high efficiency in ARIANNA. The cosmic ray flux at a mean energy of $6.5^{+1.2}_{-1.0}times10^{17}$ eV is measured to be $1.1^{+1.0}_{-0.7}times10^{-16}$ eV$^{-1}$km$^{-2}$sr$^{-1}$yr$^{-1}$ and one five-fold coincident event is used to illustrate the capabilities of the ARIANNA detector to reconstruct arrival direction and energy of air showers.
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