The internal-wave system is calculated for a body oscillating transversely, and translating uniformly, through an infinite stratified fluid of constant Brunt-Väisälä frequency. A linearized, time-dependent analysis is used, in which the vertical displacement of a fluid element is the basic dependent variable. Axisym-metric slender-body theory for a homogeneous fluid is used to determine the time-dependent source and dipole distributions required to represent the motion of the body for excitation of the internal waves. The equations are solved by Fourier-transform techniques; and the internal-wave amplitude is evaluated in the far field by the method of stationary phase. The surfaces of constant phase are found to change character as the ratio of the oscillation frequency ω of the body to the Brunt-Väisälä frequency N varies through unity. Along preferred directions, the amplitude of the internal waves is found to decay inversely with distance to the $\frac{5}{6}$ power, whereas, for uniform translation, the amplitude of the internal waves falls off inversely with distance from the body. An asymptotic expression for the amplitude in preferred directions is calculated for several values of the ratio ω/N.