Modons correspond to isolated dipole vortex solutions of the quasigeostrophic equations. They have been proposed as prototype models for some geophysical (and plasma) vortices. The classical modon solution on a &bgr; plane does not permit a Rossby wave field in the exterior or far‐field region of the modon. However, it is qualitatively known that the gravest mode associated with a normal mode decomposition of a stationary modon in a continuously stratified fluid of finite depth necessarily contains a Rossby wave tail in the downstream region if the background flow is eastward. The same effect can be formally recreated in an equivalent‐barotropic model of a stationary modon embedded in a constant eastward zonal flow. An analytical solution to this problem satisfying the correct upstream radiation condition is presented and its dynamical characteristics are discussed.