Spin-polarized atomic hydrogen is used as a gaseous polarized-proton target in high-energy and nuclear-physics experiments operating with internal beams in storage rings. When such beams are intense and bunched, this type of target can be depolarized by a resonant interaction with the transient magnetic field generated by the beam bunches. This effect has been studied with the HERA positron beam in the HERMES experiment at DESY. Resonances have been observed and a simple analytic model has been used to explain their shape and position. Operating conditions for the experiment have been found where there is no significant target depolarization due to this effect. [S0031-9007(98)08340-9].