The authors consider the problem of an ellipsoid performing simple-harmonic swaying or yawing oscillations of small amplitude in the free surface of an ideal incompressible fluid, with gravity. The problem is examined in the limit of low frequency. The first nonzero term in the radiation pattern, which is a dipole term, and the damping coefficient, are found for sway. In the case of yaw the first nonzero term in the radiation pattern is a quadrupole term, and involves a higher power of frequency. Hence, the damping coefficient for yaw is of smaller order than the damping coefficient for sway, at low frequency. The strip-theory prediction of the sway damping coefficient is compared with the exact value and is shown to be too large by a factor of order K.