Noise driven relaxation oscillations of a solid-state ring laser under
going stable dynamical periodic pulsation are investigated for the fir
st time. Formulas for the relaxation oscillation frequencies in the pr
esence of frequency nonreciprocity are derived. It is shown that, in t
he case of symmetric backscattering, the self-modulation regime is cha
racterized by damped relaxation oscillations. The conditions are found
for which the relaxation oscillation frequencies are nondegenerate ov
er the whole range in which the frequency nonreciprocity can be varied
. The dependence of the relaxation oscillation frequencies on paramete
rs of the sinusoidally alternating bidirectional ring laser is substan
tially different from the corresponding dependence for unidirectional
operation.