Prolate spheroidal coordinates and associated prolate spheroidal wave functions have been used to study the sound radiation from arbitrarily vibrating prolate spheroids. A local radiation impedance is defined and discussed. This impedance shows the effect of the strong locally variable nearfield and indicates that the spheroid generates L/2R (L/2R = fineness ratio) more acoustic power per unit area at the equator than at the poles. The high-frequency asymptotic limit of the impedance is determined and from it a radiation criterion of higher modes determined.