We present a detailed analysis of a high-bandwidth quantum memory protocol for storing single photons in a rare-earth-ion doped crystal. The basic idea is to benefit from a coherent free-induced decay type re-emission which occurs naturally when a photon with a broadband spectrum is absorbed by a narrow atomic transition in an optically dense ensemble. This allows for a high-bandwidth memory for realistic material parameters. Long storage time and on-demand readout are obtained by means of spin states in a lambda-type configuration, through the transfer of the optical coherence to a spin coherence (so-called spin-wave storage). We give explicit formulae and show numerical results which make it possible to gain insight into the dependence of the memory efficiency on the optical depth and on the width and the shape of stored photons. We present a feasibility study in rare-earth doped crystals and show that high efficiencies and high bandwidth can be obtained with realistic parameters. High-bandwidth memories using spin-wave storage offers the possibility of very high time-bandwidth products, which is important for experiments where high repetition rates are needed.