The behavior of several samples of amorphous metallic-alloy tape-wound magnetic cores excited under sine-wave voltage conditions in the 1 kHz to 100 kHz frequency range is investigated. These tape-wound cores made by Magnetics, Inc. from two different Allied Corporation amorphous metallic-glass alloys, METGLAS® alloy 2605S-3 (composition Fe79B16Si5) and METGLAS® alloy 2605SC (composition Fe81B13.5Si3.5C2), are studied along with Magnetics, Inc. F ferrite cores. Measurements of the induced-voltage and exciting-current waveforms are used to obtain dynamic (ac) core characteristic loops of flux density B versus net ampere turns per unit magnetic path length Npip/lm, as well as core loss. The behavior of one of the amorphous metallic-glass alloys (Alloy 2605S-3) at flux-density levels in the range of 0.5 T and 1 T is found to lead to core characteristic loops with strange shapes and to unusual variations in the plots of energy loss per unit volume per cycle versus frequency. The second amorphous alloy (Alloy 2605SC) shows similar unusual properties but only at high flux-density levels close to saturation, above 1.25 T. The unexpected behavior observed in these two alloys is due to magnetoelastic resonances of the toroidally shaped cores. By modeling the tape-wound core as a thin ring, calculations are made for the resonant frequencies corresponding to the mode of extensional vibrations which agree well with experimentally measured results.