Aims. We wish to study the spectral dependence of the radio emission from cosmic-ray air showers around $100~{\rm PeV}$ $(10^{17}~{\rm eV})$. Methods. We observe short radio pulses in a broad frequency band with the dipole-interferometer LOPES (LOFAR Prototype Station), which is triggered by a particle detector array named Karlsruhe Shower Core and Array Detector (KASCADE). LOFAR is the Low Frequency Array. For this analysis, 23 strong air shower events are selected using parameters from KASCADE. The radio data are digitally beam-formed before the spectra are determined by sub-band filtering and fast Fourier transformation. Results. The resulting electric field spectra fall off to higher frequencies. An average electric field spectrum is fitted with an exponential $E_{\nu}=K\cdot\mbox{exp}\,(\nu/\mathrm{MHz}/\beta)$ and $\beta=-0.017\pm0.004$, or alternatively, with a power law $\epsilon_{\nu}=K\cdot\nu^{\alpha}$ and a spectral index of $\alpha=-1\pm0.2$. The spectral slope obtained is not consistent within uncertainties and it is slightly steeper than the slope obtained from Monte Carlo simulations based on air showers simulated with CORSIKA (Cosmic Ray Simulations for KASCADE). For the analyzed sample of LOPES events, we do not find any significant dependence of the spectral slope on the electric field amplitude, the azimuth angle, the zenith angle, the curvature radius, nor on the average distance of the antennae from the shower core position. But one of the strongest events was measured during thunderstorm activity in the vicinity of LOPES and shows the longest pulse length measured of $110~\mathrm{ns}$ and a spectral slope of $\alpha=-3.6$. Conclusions. We show with two different methods that frequency spectra from air shower radio emission can be reconstructed on event-by-event basis, with only two dozen dipole antennae simultaneously over a broad range of frequencies. According to the obtained spectral slopes, the maximum power is emitted below 40 MHz. Furthermore, the decrease in power to higher frequencies indicates a loss in coherence determined by the shower disc thickness. We conclude that a broader bandwidth, larger collecting area, and longer baselines, as will be provided by LOFAR, are necessary to further investigate the relation of the coherence, pulse length, and spectral slope of cosmic ray air showers.