[en] A bunched beam that is accelerated by the fundamental resonant mode of RF cavities necessarily loses energy by exciting the higher-order modes of these very cavities. We consider here the calculation of the power lost by gaussian bunches traversing a structure with resonator impedance of arbitrary quality factor. Known formulas are used for the infinite summations, which reduce to other approximate ones for the broad-band case. For the narrow-band case we present a simple analytic expression. We apply our results to calculate the power lost to the higher-order modes of the SSC RF cavities. Our conclusion is that if the resonant frequencies of all higher-order modes of the cavities are sufficiently far from integral multiples of the bunch frequency, the energy loss is strongly reduced over that calculated by a simple broad-band model; but if this condition is not satisfied, then the resulting resonant loss for even a weak mode may be several orders of magnitude greater than the broad-band calculation. 8 refs